This is probably not a research level question, I'm sorry if it is inappropriate. I'm reasking here this question on math.se.

Suppose that $\xi: E \to B$ is a bundle (by which I mean simply a continuous mapping), and there exists a manifold $F$ such that $\Gamma(\xi) \cong C(B, F)$ as sheaves over $B$ (here by $\Gamma$ I mean the sheaf of sections, and by $C$ I mean the sheaf of continuous functions from $B$ to $F$). Does this imply that $\xi$ is locally trivial? If the proof is simple, I would like a hint on where to start, otherwise I would like a reference that studies this and related topics in more detail, please.

UPD: I guess it boils down to whether a bundle $\xi: E \to B$ such that $\Gamma(\xi) \cong C(B, \mathbb{R}^n)$ is (locally) trivial, then one can take preimages of charts to obtain the cover of $B$ that I hope can be used in the proof, but not having really worked with sheaves before I'm not sure.

UPD: Please comment if you think this quesion should be retagged, rephrased, or something like that. Having received no comments whatsoever on both math.se and MO, I don't know what to think :(