Tangent bundles of exotic manifolds

Consider a pair of homeomorphic but not diffeomorphic smooth manifolds $M_1$ and $M_2.$ Fix a homeomorphism $\phi\colon M_1\rightarrow M_2.$ If I understand correctly, two bundles $TM_1$ and $\phi^*TM_2$ are always isomorphic as topological fiber bundles (that's because they are both isomorphic to a subbundle of the tangent microbundle).

Are they always isomorphic as vector bundles?

John Milnor gives an example of two homeomorphic smooth manifolds whose tangent bundles are not isomorphic as topological vector bundles, see his ICM-1962 address, Corollary 1. I think, this was the first such example.

Edit. Few more things (motivated by questions in comments below):

1. The total space of the tangent bundle to any exotic $n$-sphere is diffeomorphic to $TS^n$, see

R. De Sapio, Disc and sphere bundles over homotopy spheres, Math. Z. 107 (1968) 232-236.

2. If $M_1, M_2$ are two homeomorphic manifolds, then the total spaces of their tangent bundles $TM_1, TM_2$ are always homeomorphic (the tangent bundles are even topologically isomorphic as microbundles), this follows from

J.Milnor, Microbundles-I, Topology, 3 (1964) 53-80.

3. I do not know of any examples where tangent bundles of two smooth homeomorphic manifolds $M_1, M_2$ such that the total spaces of $TM_1, TM_2$ are not diffeomorphic, but I did not spend much time thinking about this either.