POSNER, Chief Judge.

The parties to these consolidated appeals, an employer and a union, have a collective bargaining agreement that provides for arbitration of disputes arising under it. The employer changed certain of its staffing practices in an effort to avoid paying overtime wages. The change precipitated a slew of grievances by the union, of which seven resulted in arbitration proceedings (all before different arbitrators) that are relevant to the appeal. The issue in all the arbitrations was the same: whether the change in staffing practices violated the collective bargaining agreement. The employer won all but the fourth of the arbitrations. The fourth differs from the other six in not involving claims for compensation for individual workers adversely affected by the challenged staffing practice. The award in the fourth is a general declaration that the staffing practice violates the collective bargaining agreement. The other six sought compensation for individual workers. The union sought enforcement of the award in its favor in a federal district court under the Taft-Hartley Act, which in section 301 ( 29 U.S.C. § 185) makes collective bargaining agreements enforceable. The court enforced the award, and the employer did not appeal.

The company had won the first three arbitrations after the hearing before the fourth arbitrator but before he rendered his award in favor of the union. But neither party told the fourth arbitrator about the outcome of those arbitrations. In the case under review — the second round in the district court, the first having ended with the court's confirmation of the fourth award, the one in favor of the union — the district court confirmed the six arbitration awards that the company had won, all of which were rendered before the district court confirmed the fourth award. The union appeals.

The upshot of the two district court decisions, one enforcing the award in favor of the union and the other enforcing the awards in favor of the company, is not a happy one, since the decisions are inconsistent, with six of them rejecting the premise of the seventh (that is, of number 4). Yet it is not quite the recipe for anarchy that it may appear to be. The first district court decision, in confirming the fourth arbitrator's award, did not order the employer to rescind the change in its staffing practices (courts are reluctant to issue labor injunctions, though they will enforce an arbitrator's injunction, e.g., Local 1545 v. Inland Steel Coal Co., 876 F.2d 1288, 1292-96 (7th Cir. 1989); United Electrical, Radio Machine Workers v. Honeywell Inc., 522 F.2d 1221, 1225-28 (7th Cir. 1975); Derwin v. General Dynamics Corp., 719 F.2d 484, 491 (1st Cir. 1983), which however the fourth arbitrator's award was not), while the second decision, confirming a bunch of awards in favor of the employer, authorizes the employer to continue in its changed course with respect to the workers involved in those arbitrations but does not rescind the order of the fourth arbitrator, which would be flatly contrary to the district court's first decision. (Hence the district court was correct to reject the union's argument that by continuing to arbitrate the issue resolved in the union's favor by the fourth arbitrator, the company was in contempt of the district court's order confirming the fourth arbitrator's award.) Any subsequent disputes are likely to be resolved in the company's favor as well, as we shall see; and the company points out that in any event, since judicial review of an arbitration award is so limited as to be little better than a rubber stamp, arbitral awards based on diametrically opposed interpretations of the identical contract can all withstand judicial review. E.g., American Nat'l Can Co. v. United Steelworkers of America, 120 F.3d 886, 891-93 (8th Cir. 1997); Connecticut Light Power Co. v. Local 420, 718 F.2d 14, 20-21 (2d Cir. 1983); Westinghouse Elevators of Puerto Rico, Inc. v. S.I.U. de Puerto Rico, 583 F.2d 1184 (1st Cir. 1978). This is provided that the inconsistent interpretations do not result in the kind of impasse that would be presented if the employer were simultaneously enjoined by a court from continuing, and authorized by the court to continue, the identical practice. But, as we have seen, that is not a problem here.

Yet when multiple suits as closely related as are these seven arbitrations are filed in court, they invariably are consolidated. However, the consolidation of arbitrations, as distinct from the consolidation of court suits, is a creature of contract, Connecticut Gen'l Life Ins. Co. v. Sun Life Assurance Co., No. 99-4085, 2000 WL 490692, *2 (7th Cir. April 27, 2000), and the collective bargaining agreement in this case permits consolidation of grievances (the filing of a grievance being the first stage in the dispute resolution process that includes, as its final stage before judicial review, the proceeding before the arbitrator) only if both union and employer agree to consolidate them, and they did not do so here. Nevertheless, when, before the fourth arbitrator ruled, the first three arbitrations came out in favor of the employer, one might have expected the employer to so advise the fourth arbitrator, and this it failed to do. It argues that it could not do so because the collective bargaining agreement forbids reopening the record after the hearing before the arbitrator. The argument is unsound. The decisions of the other three arbitrators were not evidence — the sort of thing a prohibition against reopening the record would cover. They were the arbitral equivalents of judicial decisions, of which, of course, a court can take judicial notice; and in just the same way an arbitrator can take "arbitral notice" of a previous decision by another arbitrator dealing with the same or a related issue — especially since, as we are about to see, the collective bargaining agreement requires arbitrators to give prior arbitral decisions preclusive effect, which they cannot do if no one informs them of those decisions.

That effect, like most features of arbitration, is indeed a matter of contract rather than a matter of law. E.g., Pierce v. Commonwealth Edison Co., 112 F.3d 893, 895-96 (7th Cir. 1997); Brotherhood of Maintenance of Way Employees v. Burlington Northern R.R., 24 F.3d 937, 940 (7th Cir. 1994); American Nat'l Can Co. v. United Steelworkers of America, supra, 120 F.3d at 891-92; Hotel Ass'n of Washington, D.C., Inc. v. Hotel Restaurant Employees Union, 963 F.2d 388 (D.C. Cir. 1992); District 37 v. Lockheed Engineering Mgmt. Services Co., 897 F.2d 768, 773 (5th Cir. 1990); G. Richard Shell, "Res Judicata and Collateral Estoppel Effects of Commercial Arbitration," 35 UCLA L. Rev. 623, 660-63 (1988). If the parties to the collective bargaining agreement want the first arbitrator's interpretation of a provision of the agreement or resolution of a dispute arising under the agreement to have preclusive effect, they can so provide; and whether they do so or not, the question of the preclusive force of the first arbitration is, like any other defense, itself an issue for a subsequent arbitrator to decide. E.g., Independent Lift Truck Builders Union v. NACCO Materials Handling Group, Inc., 202 F.3d 965, 968 (7th Cir. 2000); Brotherhood of Maintenance of Way Employees v. Burlington Northern R.R., supra, 24 F.3d at 940; Chiron Corp. v. Ortho Diagnostic Systems, Inc., 207 F.3d 1126, 1132-33 (9th Cir. 2000); John Hancock Mutual Life Ins. Co. v. Olick, 151 F.3d 132, 139-40 (3d Cir. 1998); National Union Fire Ins. Co. v. Belco Petroleum Corp., 88 F.3d 129, 135-36 (2d Cir. 1996). Most arbitrators, applying the arbitral equivalent of common law, would indeed give preclusive effect to the first arbitration, even in the absence of contractual guidance, provided that the usual conditions of res judicata were fulfilled. See Fairweather's Practice Procedure in Labor Arbitration 525-33 (4th ed., Ray J. Schoonhoven ed. 1999); Timothy J. Heinsz, "Grieve It Again: Of Stare Decisis, Res Judicata and Collateral Estoppel in Labor Arbitration," 38 B.C. L. Rev. 275, 286-93 (1997); Shell, supra, 35 UCLA L. Rev. at 663-64. Anyway, here the contract did make clear that res judicata is a defense in a subsequent arbitration. See Heinsz, supra, 38 B.C. L. Rev. at 286-87, describing the contract. So by failing to argue preclusion to the fourth arbitrator, the employer forfeited a good defense that would have headed off the confusion that ensued.

It might seem that once the fourth arbitrator's decision came down, res judicata clicked in and entitled the union to block any subsequent inconsistent arbitration awards unless and until that decision was vacated by a reviewing court. Not so. It was up to the subsequent arbitrators, 5 through 7, to decide what significance to attach to the employer's having booted its res judicata defense before the fourth arbitrator; and they decided to give it no weight. They forgave the employer's waiver, and like other procedural questions in arbitration not regulated by law their determination as to whether to forgive the waiver cannot be second-guessed by a court. Connecticut Gen'l Life Ins. Co. v. Sun Life Assurance Co., supra, at *2; Baravati v. Josephthal, Lyon Ross, Inc., 28 F.3d 704, 709 (7th Cir. 1994); UHC Management Co. v. Computer Sciences Corp., 148 F.3d 992, 997 (8th Cir. 1998). They did it, oddly, out of a commitment to res judicata so profound as not to brook a forfeiture: they held that they were bound by the decisions made by the first three arbitrators.

It doesn't follow that it was right for the district court in the second go round to enforce the other arbitration awards (that is, 1 through 3 and 5 through 7). We must consider the possible res judicata effect of the first district court decision, a distinct issue from the res judicata effect of the arbitral awards. The union argues that the district court's judgment in the first round, when the issue before the court was whether to enforce the fourth arbitrator's award, which was in favor of the union, was res judicata. The other six awards, the ones in favor of the employer, all were rendered before the district court confirmed the fourth award. The employer's claim for judicial confirmation of those awards was, the union argues, a compulsory counterclaim to the union's claim for enforcement of the fourth arbitrator's award, because both claims arose out of the same transaction or occurrence, Fed.R.Civ.P. 13(a), and failure to plead a compulsory counterclaim bars as a matter of res judicata its being presented in a subsequent suit. Baker v. Gold Seal Liquors, Inc., 417 U.S. 467, 469 n. 1 (1974); Publicis Communication v. True North Communications Inc., 132 F.3d 363, 365 (7th Cir. 1997); Olympia Hotels Corp. v. Johnson Wax Development Corp., 908 F.2d 1363, 1367 (7th Cir. 1990); Driver Music Co. v. Commercial Union Ins. Cos., 94 F.3d 1428, 1435 (10th Cir. 1996); 6 Charles Alan Wright, Arthur R. Miller Mary Kay Kane, Federal Practice Procedure § 1417, p. 129 (2d ed. 1990).

But while the seven arbitrations all arose out of the same transaction or occurrence, namely the staffing dispute, the two district court proceedings did not. They arose out of the arbitrations, which we deem to have been separate transactions or occurrences. We make this classification not under the compulsion of the words of Rule 13(a), but because of the need to respect the right of the arbitrators to resolve disputes within the scope of the authority granted them by the collective bargaining agreement. The arbitrators who administer this agreement have, we noted, a fierce commitment to res judicata. Paradoxically, it would be nullified were we to display an equal zeal for the principle of res judicata embedded in Rule 13(a) — if, that is to say, we agreed with the union that the district court was precluded from confirming the arbitral awards that were based on the arbitrators' commitment to arbitral res judicata. And, precisely because of their commitment, we needn't worry overmuch that the district court will be asked to confirm inconsistent awards from now until every single worker at Consolidation Coal Company's plant has gone through arbitration. Given the rule of res judicata applied by the arbitrators under this collective bargaining agreement, it is reasonably certain that all subsequent arbitrations will be decided in favor of the company and will be confirmed by the district court if the union refuses to bow to them. The essential point, however, is that the res judicata effect of a judicial decision merely confirming an arbitral award is extremely limited. All it amounts to is a determination that there is no basis for upending that award; the effect on subsequent awards must be left to the arbitrators who make them.

The union has no colorable defense to the confirmation of the awards in favor of the company other than res judicata, and so the district court's judgment is

Affirmed.