@swamidass and @dga471, I read up on the relativistic field theory generalization of the pilot-wave mechanics (analogous to Orthodox QM -> Orthodox QFT), and it turns out that the pilot-wave interpretation is no longer deterministic in this wider picture. Interestingly, it seems that people who are in favor of the pilot-wave interpretation are not perturbed by this despite the fact that it destroys what I think is the main draw towards the pilot wave interpretation.
What follows is my summary. To non-physicists reading this post: unfortunately it’s hard to talk about this subject without a little bit of jargon, so forgive my somewhat liberal use of them. I did not use any jargon that is silly enough to not have their own wikipedia page, so you can consult wiki for the meaning of specific terms.
The gist of the issue is that the pilot-wave interpretation puts a special emphasis on the position basis. In my previous post, I call this a choice of a preferred frame/foliation of spacetime. Now, in QFT the position basis is not well defined. This just comes from the fact that states in QFT have a variable number of particles, so it does not make sense to have a state where there exists # number of particles at (x1, x2, x3, …). Because of this, there is a huge disconnect between the pilot-wave’s special emphasis of the particle position and the non-existence of the position basis in QFT.
The way this is remedied is by setting up a large, time-dependent configuration space of possible positions for a variable number of particles, which I will call Q.
Think of the configuration space of one particle as a box where position in the box labels the position of a particle; call this C1. Suppose there are two particles, this is a higher dimensional box (3 spatial dimension * 2 particles = 6 dimensional box), and the position in this higher dimensional box corresponds to the positions of the two particles; call this C2. Since particles in a QFT can be created and destroyed, the configuration space Q must include both C1 and C2. Indeed, Q must include not only C1 and C2 but everything up to CInfinity. So, Q is a disjoint union of all the n-particle configuration spaces.
If you think this is a super clunky configuration space, you would be correct. When the path of the system is within a single Cx, the path is given by the pilot-wave equation deterministically. However, the jumps between Cx and Cy (remember these are disjoint sets within Q) is completely stochastic in both when the jump happens and where the jump leads to in the next hyperbox.
The main point is that once relativity and thus field-theory is taken into account, the pilot-wave interpretation is not deterministic. Since in the real world relativity is true, I think a more punchy statement can be said: In the real world, the pilot-wave interpretation is not deterministic.
Edit: grammar