The main issue is that QFT does not generate a mathematically well-defined evolution of the wave function, but there are promising directions on that front as well: http://arxiv.org/abs/1506.00497
Basically, they treat the Hamiltonian as a whole instead of a perturbation and use a domain for it that has the probabilities flowing from one area to another appropriately to the different particle sectors.
There is also work on dealing with relativity which is a separate issue. While there are solutions, mainly centering on finding a natural foliation, the tension with relativity has not really been resolved, a tension which exists regardless of interpretation.
Exactly, John Bell had it right 50 years ago: non-locality is the problem of QM that needs to be solved, and other interpretations let you simply gloss over the problem without really addressing it. The recent work for finding a natural foliation present in any formalism with a wave function, including Bohmian mechanics, proves every QM shares this problem (or if you think one doesn't, then none of them do for the exact same reasons).