  The surfaces considered are real, rational and have a unique smooth real
$(-2)$-curve. Their canonical class $K$ is strictly negative on any other
irreducible curve in the surface and $K^2>0$. For surfaces satisfying these
assumptions, we suggest a certain signed count of real rational curves that
belong to a given divisor class and are simply tangent to the $(-2)$-curve at
each intersection point. We prove that this count provides a number which
depends neither on the point constraints nor on deformation of the surface
preserving the real structure and the $(-2)$-curve.
