Let $k=\overline{\mathbf{Q}}$. Fix a finite closed subset $B\subset \mathbf{P}^1_k$. Let $X$ be a "nice" topological space and suppose that there is a continuous morphism $f:X\to \mathbf{P}^1_k-B$.
Assume that $f$ is a covering space topologically of degree $d<\infty$.
Can we give $X$ the structure of a smooth affine curve such that $f:X\to \mathbf{P}^1_k-B$ becomes a finite etale morphism?
This is possible if $k=\mathbf{C}$.