  Let $X\rightarrow C$ be a totally real semistable degeneration over a smooth
real curve $C$ with degenerate fiber $X_0$. Assuming that the irreducible
components of $X_0$ are simple from a cohomological point of view, we give a
bound for the individual Betti numbers of a real smooth fiber near $0$ in terms
of the complex geometry of the degeneration. This generalizes previous work of
Renaudineau-Shaw, obtained via combinatorial techniques, for tropical
degenerations of hypersurfaces in smooth toric varieties. The main new
ingredient is the use of real logarithmic geometry, which allows to work with
not necessarily toric degenerations.
