  For a real K3-surface $X$, one can introduce areas of connected components of
the real point set $\mathbb{R} X$ of $X$ using a holomorphic symplectic form of
$X$. These areas are defined up to simultaneous multiplication by a positive
real number, so the areas of different components can be compared. In
particular, it turns out that the area of a non-spherical component of
$\mathbb{R} X$ is always greater than the area of any spherical component.
  In this paper we explore further comparative restrictions on the area for
real K3-surfaces admitting a suitable polarization of degree $2g - 2$ (where
$g$ is a positive integer) and such that $\mathbb{R} X$ has one non-spherical
component and at least $g$ spherical components. For this purpose we introduce
and study the notion of simple Harnack curves in real K3-surfaces, generalizing
planar simple Harnack curves.
