Empty real Enriques surfaces and Enriques-Einstein-Hitchin 4-manifolds

We prove that the moduli space of empty real Enriques surfaces (and, thus, the moduli space of compact orientable 4-dimensional Einstein manifolds whose universal covering is a K3-surface and \pi_1(E) = Z/2 x Z/2) is connected. The proof is based on a systematic study of real elliptic pencils and gives explicit models of all empty real Enriques surfaces.