  Classification of real K3 surfaces X with a non-symplectic involution \tau is
considered. For some exactly defined and one of the weakest possible type of
degeneration (giving the very reach discriminant), we show that the connected
component of their moduli is defined by the isomorphism class of the action of
\tau and the anti-holomorphic involution \phi in the homology lattice. (There
are very few similar cases known.) For their classification we apply invariants
of integral lattice involutions with conditions which were developed by the
first author in 1983. As a particular case, we describe connected components of
moduli of real non-singular curves A\in |-2K_V| for the classical real
surfaces: V=P^2, hyperboloid, ellipsoid, F_1, F_4.
  As an application, we describe all real polarized K3 surfaces which are
deformations of general real K3 double rational scrolls (surfaces V above).
There are very few exceptions. For example, any non-singular real quartic in
P^3 can be constructed in this way.
