Classification of real K3 surfaces $X$ with non-symplectic involution $\tau$ is considered. We define a natural notion of degeneration for them. We show that the connected component of moduli of non-degenerate surfaces of this type is defined by the isomorphism class of the action of $\tau$ and the anti-holomorphic involution $\varphi$ in the homology lattice. (There are very few similar results known.) For their classification we apply invariants of integral lattice involutions with conditions that were developed by the first author in 1983. As a particular case, we describe connected components of moduli of real non-singular curves $A \in | -2 K_V|$ for the classical real surfaces: $V = P^2$, hyperboloid, ellipsoid, $F_1$, $F_4$.

As an application, we describe all real polarized K3 surfaces that are deformations of general real K3 double rational scrolls (the surfaces $V$ above). There are very few exceptions. For example, any non-singular real quartic in $P^3$ can be constructed in this way.