Bishop-Runge approximations and inversion of a Riemann-Klein theorem

In this paper we give results about projective embeddings of Riemann surfaces, smooth or nodal, which we apply to the inverse Dirichlet-to-Neumann problem and to the inversion of a Riemann-Klein theorem. To produce useful embeddings, we adapt a technique of Bishop in the open bordered case and use Runge type harmonic approximation theorem in the compact case.