Holomorphic maps sharing preimages over finitely generated fields

Let $R$ be a compact Riemann surface, and let $P: R \to \mathbb P^1(\mathbb C)$ and $Q: R \to \mathbb P^1(\mathbb C)$ be holomorphic maps. In this paper, we investigate the following problem: under what conditions do the preimages $P^{-1}(K)$ and $Q^{-1}(K)$ coincide for some infinite set $K$ contained in $\mathbb P^1(k)$, where $k$ is a finitely generated subfield of $\mathbb C$ (e.g., a number field)? Equivalently, we study holomorphic correspondences that admit an infinite completely invariant set contained in $\mathbb P^1(k)$. We show that if such a set exists, then there is a holomorphic Galois covering $\Theta: R_0 \to \mathbb P^1(\mathbb C)$, where $R_0$ has genus zero or one, such that $P$ and $Q$ are ``compositional left factors" of $\Theta.$ We also consider a more general equation $P^{-1}(K_1) = Q^{-1}(K_2),$ where $K_1$ and $K_2$ are infinite subsets of $\mathbb P^1(k)$.