We study the plasma-vacuum interface problem in relativistic magnetohydrodynamics for the case when the plasma density does not go to zero continuously, but jumps. Unlike the nonrelativistic version of this problem, due to the presence of a non-zero displacement current in vacuum, the planar interface can be violently unstable. By using a suitable secondary symmetrization of the Maxwell equations in vacuum, we find a sufficient condition that precludes violent instabilities. We prove the local-in-time existence and uniqueness of smooth solutions of the original nonlinear free boundary problem provided that this neutral stability condition is satisfied at each point of the initial interface.