Complex Lie inverse Galois problem

Question

My question is about the inverse Galois problem for infinite dimensional complex manifolds. If $K$ is the field of meromorphic functions over a complex manifold $M$ and $G$ is a finite or infinite dimensional complex Lie group, have we a fibration $f: N \rightarrow M$, perhaps not a complex fiber bundle, such that $G$ is the complex group of all the biholomorphic maps of $N$ onto $N$ with fixed fibration $f$, or the group of automorphisms of the meromorphic functions of $N$ with $K$, the ones from $M$, fixed? ($G$ is the gauge group of the fibration.)