In Miranda's book on algebraic curves and Riemann surfaces, Miranda writes:
It is a basic and highly nontrivial result that a compact Riemann surface has nonconstant meromorphic functions on it [...] The theory involved in producing meromorphic functions for an unknown compact Riemann surface is rather technical analysis and functional analysis. After one has access to meromorphic functions, however, the theory is completely algebraic, or at least can be made to be so.
I've seen this claim a number of other places as well. It seems unnatural to use real analysis to prove a theorem about Riemann surfaces, which are geometric/algebraic objects. Is there genuinely no purely geometric/algebraic way to realize an abstract Riemann surface as a branched cover of the Riemann sphere?