On cohomology classes defined by the real points of a real algebraic $\operatorname{GM}$-surface

The cohomology classes $x_i=[X_i]^*\in H^2(X(\mathbb C),\mathbb Z)$ are studied, where $X_1,\dots,X_m$ are the connected components of the set of real points $X(\mathbb R)$ of a real algebraic $\operatorname{GM}$-surface $X$ and $X(\mathbb R)=X_1\cup\dots\cup X_m$ is assumed to be orientable. The results are applied to obtain congruences for the Euler characteristic of $X(\mathbb R)$.