Bohmian Mechanics fails to compute multi-time correlations

The violation of Bell type inequalities in quantum systems manifests that quantum states cannot be described by classical probability distributions. Yet, Bohmian mechanics is a realistic, non-local theory of classical particle trajectories that is claimed to be indistinguishable by observations from more traditional approaches to quantum mechanics. We set up a spatial version of the GHZ system with qubits realised as positional observables that demonstrates that the Bohmian theory fails to match predictions of textbook quantum mechanics (and most likely experients) unless enlarged by a microscopic theory of collapse of the wave function after observation. For this discrepancy to occur it is essential that positions at different times do not commute.