We study qualitative aspects of the Welschinger-like $\mathbb Z$-valued count
of real rational curves on primitively polarized real $K3$ surfaces. In
particular, we prove that with respect to the degree of the polarization, at
logarithmic scale, the rate of growth of the number of such real rational
curves is, up to a constant factor, the rate of growth of the number of complex
rational curves. We indicate a few instances when the lower bound for the
number of real rational curves provided by our count is sharp. In addition, we
exhibit various congruences between real and complex counts.Comment: corrected typos, accepted for publicatio