For any such $K$, we can recover the subfield of constants $\mathbb{C}\subset K$ as the elements of $K$ that have $n$th roots for all $n$.  Indeed, if a meromorphic function on a compact Riemann surface has roots of all orders, it must have valuation $0$ at every point and hence be constant.  Thus if $K$ and $K'$ are isomorphic as abstract fields, they are also isomorphic over $\mathbb{C}$ (which happens iff the corresponding Riemann surfaces are isomorphic).