I am very confused about the base affine space. Let $G$ be a complex reductive algebraic group and $H=B/U$ the abstract torus.

If I understand it correctly (Edit: which turns out not to be the case! See Sams answer), the base affine space is an $H$ bundle on the flag variety, such that every fiber is canonically isomorphic to $H$? Now common sense tells that a bundle where all fibers are canonically the same should be trivial, right !?

On the other hand, I think I calculated that if base affine space was trivial, all the interesting TDOs $\mathcal D_\chi$ for $\chi \in \mathfrak h^* $ on the flag variety would be isomorphic...

So my questions are:

Is base affine space trivial? If not, why does common sense fail here?

$\textbf{Bonus question:}$ Can you give (other) examples of nontrivial bundles with all fibers canonically isomorphic?