Let phi : k --> A and f : A --> R be ring morphisms, R a real ring. We prove that if f : A --> R is etale, then the corresponding mapping between real Riemann surfaces S-r(f) : S-r(R/k) --> S-r(A/k) is a local homeomorphism. Several preparatory results are proved, as well. The most relevant among these are: (1) a Chevalley's theorem for real Riemann surfaces on the preservation of constructibility via S-r(f), and (2) an analysis of the closure operator on real Riemann surfaces. Constructible sets are dealt with by means of a suitable first-order language.