Bohmian mechanics has a very elegant way of dealing with identical particles. I know as I am a coauthor on such papers. The idea is that instead of R^3N with its labels, we use a space with no labels on particles. The only relevant data are the locations of the particles, not some extrinsic numbering. This immediately gives us bosons. Fermions take a little more work, essentially using a vector bundle approach analogous to Mobius strips. One can then prove that in 3 dimensions and for scalar wave functions, that is all the possibilities. For 2D, you also get anyons; this is all about the topology of the natural configuration space.

For spin, one has to realize that it is modeled well by wave functions that take on vectors as values. When moving to multiple particles with spin, we tensor (kind of multiply) those vectors together. But the key point is that we tensor based on the spatial location. So instead of tensoring over particle 1 and particle 2, we tensor over their positions. This gives us complicated vector bundles. It opens a lot more possibilities, but when you look at general quantum dynamics, the only stable ones in 3D are the bosons and fermions.

Thus, not only can Bohmian mechanics account for it, it actually provides a very natural explanation. It is completely rooted in considering what the configuration space is. Where Bohmian mechanics shines is that what the configuration space is comes from what the theory itself is concerned with, namely particles with definite positions.

And just as a side note, the standard formalism of quantum mechanics emerges in a mathematically rigorous way from Bohmian mechanics. Everything in non-relativistic quantum mechanics is completely accounted for and you get a clear foundation for extending it to other spaces and more general questions.

As for relativity and quantum field theory, that is still a work in progress, but the main stumbling block is actually having a well-defined evolution of the wave function in interacting cases. If we had that, then the Bohmian additions are easily handled for the most part. Research has been done and is being done.

A simple (layperson) reference for this is sadly lacking, but you can try either http://arxiv.org/pdf/quant-ph/0506173.pdf or my thesis (chapter 5) at http://jostylr.com/thesis.pdf Hopefully they are somewhat accessible to this audience.