Quantum mechanics obeys the principle of relativistic causality (no superluminal signalling), but it is nonlocal in at least two ways: the Aharonov-Bohm effect shows that a charged particle can detect an electromagnetic field at a distance, and Bell’s theorem shows that quantum correlations are nonlocal. Could quantum mechanics be the only theory that reconciles relativistic causality and nonlocality? That is, can we derive quantum mechanics from the two axioms of relativistic causality and nonlocality? The PR box provides a counterexample, showing that quantum correlations are not the most nonlocal correlations consistent with relativistic causality. Let us consider supplementing these two axioms with a minimal third axiom: there exists a classical limit in which macroscopic observables commute. That is, just as quantum mechanics has a classical limit, so must any generalization of quantum mechanics. In this classical limit, PR-box correlations violate relativistic causality. Generalized to all stronger-than-quantum bipartite correlations, this result is a derivation of Tsirelson's bound without using quantum mechanics.