Gauge cosmic strings and large gauge transformations

I’ve been going around in circles (hah) about how gauged cosmic strings work (I’ve been using Preskill’s notes for the most part). The global string scenario makes sense to me, since different points on the vacuum manifold are physically distinct. But I’m uncertain about vacua which would seem naively to be indistinguishable since they are related by gauge transformations.

By this I mean: if you write the Higgs vev around a spatial loop as $langle phi rangle = v e^{i theta}$ or similar why can’t you just gauge away the $theta$ with a group element at every point?

I think I have worked out a conceptual explanation for this, and I’d really appreciate hearing peoples’ thoughts on whether it makes sense.

The key idea which I was not considering before is the fact that we are not identifying two vacua at a single point, but two (global) configurations of the higgs field $phi$, and we are allowed to identify them only if there exists an element of the gauge group as in the group of smooth single-valued maps from space into the group (not the group itself) which will take us from one to the other. Not only that, but the element must be a member of the homotopy class which is smoothly deformable to the identity, since elements of other winding number sectors (‘large gauge tranformations’) are not gauge transformations between physically equivalent states.

So for example, in the Abelian Higgs model, we cannot identify the configuration $langle phi(theta) rangle = v e^{i theta}$ with $langle phi(theta) rangle = v$ because the gauge transformation would have a non-trivial winding number itself.

And in the Alice $O(2)$ string model, we cannot identify $langle phi(theta) rangle = e^{i theta/2 T_3}phi(0) e^{-i theta/2 T_3}$ with $phi(theta) = phi(0)$ because the gauge transformation is either double valued or discontinuous at $theta = 2pi$.

All of these points have been mentioned in various places, including this site, but I would be grateful for confirmation that I have put them together properly.

The other thing I’d love more info on is the theory behind why large gauge transformations are not redundancies. Is it just that they are observed to not be, e.g. in the Aharonov-Bohm effect? I’ve seen ‘inequivalent quantisations’ mention (e.g. in ACuriousMind’s answer here) but I’m really not sure what that means.