I am curious about stacky generalizations of the following GAGA theorem:

If $X, U$ are complex algebraic varieties of finite type, $X$ is proper and $f:X\to U$ is an analytic map then $f$ is algebraic.

There is an established theory of analytic stacks (as well as higher analytic stacks). I am curious about the following question: for what *stacks* $U$ does the above theorem still hold (with $X$ still assumed to be a proper scheme). One case that is known to hold since the original GAGA is $U = B\mathbb{G}_m$ and I believe it is also true more general affine reductive groups. I'm interested in whether this holds for more exotic stacks, for example $BA$ for $A$ an abelian variety.

In this special case (which I am most curious about) the question can be formulated more classically: suppose $X$ is a proper scheme (say, a curve), $A$ is a polarized abelian variety, and $\mathcal{A}\to X$ is a complex-analytic principal $A$-bundle over $X$. Is the total space $\mathcal{A}$ also algebraic?