  We construct several rigid (i.e., unique in their deformation class) surfaces
which have particular behavior with respect to real structures: in one example
the surface has no any real structure, in the other one it has a unique real
structure and this structure is not maximal with respect to the Smith-Thom
inequality. So, it answers in negative to the following problems: existence of
real surfaces in each complex deformation class and existence of maximal
surfaces in each complex deformation class containing real surfaces. Besides,
we prove that there is no real surfaces among the surfaces of general type with
$p_g=q=0$ and $K^2=9$.
  As a by-product, the surfaces constructed give one more counterexample to
"Dif=Def" problem.
