A century after the advent of quantum mechanics and general relativity, both theories enjoy incredible empirical success, constituting the cornerstones of modern physics. Yet, paradoxically, they suffer from deep-rooted, so-far intractable, conflicts. Motivations for violations of the notion of relativistic locality include the Bell’s inequalities for hidden variable theories, the cosmological horizon problem, and Lorentz-violating approaches to quantum geometrodynamics, such as Horava–Lifshitz gravity. Here, we explore a recent proposal for a “real ensemble” non-local description of quantum mechanics, in which “particles” (...) can copy each others’ observable values AND phases, independent of their spatial separation. We first specify the exact theory, ensuring that it is consistent and has quantum mechanics as a fixed point, where all particles with the same values for a given observable have the same phases. We then study the stability of this fixed point numerically, and analytically, for simple models. We provide evidence that most systems are locally stable to small deviations from quantum mechanics, and furthermore, the phase variance per value of the observable, as well as systematic deviations from quantum mechanics, decay as \ time)\, where \. Interestingly, this convergence is controlled by the absolute value of energy, suggesting a possible connection to gravitational physics. Finally, we discuss different issues related to this theory, as well as potential novel applications for the spectrum of primordial cosmological perturbations and the cosmological constant problem. (shrink)