Textbooks often claim that phenomena like the Aharonov-Bohm effect require that any local formulation of quantum gauge theory use gauge potential fields. (It's also sometimes claimed that the A-B effect demonstrates that these gauge fields have physical reality in quantum gauge theory, but are just useful calculational tools in classical gauge theory. But those are philosophical waters that I don't want to wade into for this question.)
However, Lev Vaidman disputes this claim in the paper https://journals.aps.org/pra/abstract/10.1103/PhysRevA.86.040101 (arXiv:1110.6169) . He argues that if we treat the sources that generate the electromagnetic fields quantum-mechanically and consider the back-reaction that the test particle exerts on them, then we can explain the A-B effect in a completely local manner that only references the gauge-invariant electromagnetic fields. (At least, within interpretations of quantum mechanics that allow macroscopic objects to exist in superpositions.) He therefore disputes that gauge fields are necessary for explaining the A-B effect.
If we consider the possibility of magnetic monopoles, then can the Dirac quantization condition be derived within Vaidman's framework? The derivations of the DQC that I know of use topological considerations to argue that the gauge field can only be well-behaved everywhere for magnetic monopoles that obey the DQC. I can't see why the magnetic charge of magnetic monopoles would be quantized within Vaidman's framework.
Does this mean that if magnetic monopoles exist, then Vaidman's theory is physically distinct from the usual formulation of QED with magnetic monopoles in terms of nontrivial $U(1)$ fiber bundles?