Rational real algebraic models of topological surfaces

Comessatti proved that the set of real points of a rational real algebraic surface is either a nonorientable surface, or the two-sphere, or the torus. Conversely, it is easy to see that all of these surfaces admit a rational real algebraic model. We prove that they admit exactly one rational real algebraic model. This was known earlier only for the two-sphere, torus, projective plane and the Klein bottle.