Let $p: E\to B$ be a covering map of $C^\infty$ manifolds, where $E$ has a complex structure. There are many cases when we want to know whether $B$ has a complex structure (which is obviously unique) making $p$ an analytic map, for example in the construction of families of elliptic curves.

The difficulty is that given a small open set $U\subset B$, and pulling back the covering map to $$p|_{p^{-1}(U)}: \bigsqcup_\alpha U_\alpha\to U,$$ the $U_\alpha$ may induce incompatible complex structures on $U$. The easiest example of this phenomenon is the covering map $\mathbb{CP}^1\simeq S^2\to \mathbb{RP}^2$, where $\mathbb{RP}^2$ does not admit *any* complex structure, as it is not orientable. This case is not too badly behaved, however--in particular, given $U_\alpha, U_\beta\subset \mathbb{CP}^1$ over some $U\subset \mathbb{RP}^2$, the transition map $U_\alpha\to U\to U_\beta$ seems to me to be antiholomorphic.

So I have two questions about this general situation:

1) Is there an example of a covering map $p: E\to B$ of $C^\infty$ manifolds with $E$ complex, such that $B$ admits

somecomplex structure, but none making $p$ analytic?2) Given a covering map $p: E\to B$ with $E$ complex, is there an algebra-topological obstruction to the existence of a complex structure on $B$ making $p$ analytic?