Every discussion I've ever seen of the Aharonov-Bohm effect makes a big deal of its being a quantum effect with no classical analogue. But as far as I can tell it is present already at the classical level in QED. It also seems to have a close analogue in GR: the Riemann curvature outside an infinite straight cosmic string is identically zero, but an interferometer encircling it will see a phase shift that depends on its mass density.
Is there something I'm missing? If not, can someone point me to a reasonably trustworthy textbook or paper that makes the point that it's classical, especially one that also mentions the GR analogy?
(edit: By "present at the classical level" I mean that if you take the QED Lagrangian and derive classical equations of motion from it, you get a classical theory of Maxwell's electromagnetism coupled to a charged wave in which the A-B effect apparently exists just as in QED. This theory was never investigated before the quantum era, but it could have been, and the A-B effect could have been found then, as far as I can tell.
I'm hoping for a published paper by a well-known author that says the above explicitly, in part because I'd like to add it to Wikipedia.
I'm not interested in attempts to explain away the effect as being due to an external electric field, at least not for the purposes of this question. This is a theory question and there's no electric field in theory.)