  The aim of this paper is twofold. First of all, we confirm a few basic
criteria of the finiteness of real forms of a given smooth complex projective
variety, in terms of the Galois cohomology set of the discrete part of the
automorphism group, the cone conjecture and the topological entropy. We then
apply them to show that a smooth complex projective surface has at most
finitely many non-isomorphic real forms unless it is either rational or a
non-minimal surface birational to either a K3 surface or an Enriques surface.
In the second part of the paper, we construct an Enriques surface whose blow-up
at one point admits infinitely many non-isomorphic real forms. This answers a
question of Kondo to us and also shows the three exceptional cases really
occur.
