Beauville surfaces without real structures, I

Inspired by a construction by Arnaud Beauville of a surface of general type with $K^2 = 8, p_g =0$, the second author defined the Beauville surfaces as the surfaces which are rigid, i.e., they have no nontrivial deformation, and admit un unramified covering which is isomorphic to a product of curves of genus at least 2. In this case the moduli space of surfaces homeomorphic to the given surface consists either of a unique real point, or of a pair of complex conjugate points corresponding to complex conjugate surfaces. It may also happen that a Beauville surface is biholomorphic to its complex conjugate surface, neverless it fails to admit a real structure. First aim of this note is to provide series of concrete examples of the second situation, respectively of the third. Second aim is to introduce a wider audience, especially group theorists, to the problem of classification of such surfaces, especially with regard to the problem of existence of real structures on them.