So, I have heard GAGA works for Rigid Analytic spaces. I know next to nothing about this, but it made me curious as to whether there are any other contexts in which GAGA "works". Of course, this is a very vague question. Here is a way to make it more formal:
When is there a functor $F$ from the category of varieties over some field K to the category of locally ringed (in $K$-algebras) spaces. Where, say, each $F(V)$ is a locally compact Hausdorff space, and F takes proper morphisms [in the sense of algebraic geometry] to proper maps [in the topological sense]. And also that for any variety $V$, there is a morphism of locally ringed spaces $F(V) \rightarrow V$, and if $V$ is a projective variety, than this morphism induces an isomorphism between the category of coherent sheaves on $V$ and the category of coherent sheaves on $F(V)$.
Of course, these are very strong conditions, I would also be interested in weaker cases.