What precisely is in need of explanation in the Bohm-Aharanov effect?

1. The magnetic field inside the solenoid affects the particle outside the solenoid, even though the magnetic field is not outside the solenoid. Something here cannot act there.

OR

2. The invariant fact of the field is non-local; it becomes manifest only in holonomies. There is something mysterious about this since we expect interactions to occur locally, e.g. a non-zero A field here interacts with the particle here. But a suitable choice of gauge can make the A field vanish at any point.

What does it take to explain things like this?

To explain the (apparent) anomaly, the explanation must respond directly to what needs to be explained. For example:

For 1.: We must show that (a) somehow the magnetic field does exist outside the solenoid after all (true--the non-zero holonomies show this) or (b) there is nothing mysterious about something here acting there (presumably also true--it is how action at a distance was supposed to work!).

For 2.: We must show that there is nothing mysterious about the global character of the field (as manifested in the non-vanishing holonomies) having an effect. I would do this by analogy to the examples of the paper cone, which is locally flat everywhere by still bends the path of an ant. The fibre bundle representation of the effect makes this analogy a strong one.

What examples are there of this sort of explanation?

No Bell telephone theorems in the context of EPR paradox.