  We prove a bound conjectured by Itenberg on the Betti numbers of real
algebraic hypersurfaces near non-singular tropical limits. These bounds are
given in terms of the Hodge numbers of the complexification. To prove the
conjecture we introduce a real variant of tropical homology and define a
filtration on the corresponding chain complex inspired by Kalinin's filtration.
The spectral sequence associated to this filtration converges to the homology
groups of the real algebraic variety and we show that the terms of the first
page are tropical homology groups with $\mathbb{Z}_2$-coefficients. The
dimensions of these homology groups correspond to the Hodge numbers of complex
projective hypersurfaces. The bounds on the Betti numbers of the real part
follow, as well as a criterion to obtain a maximal variety. We also generalise
a known formula relating the signature of the complex hypersurface and the
Euler characteristic of the real algebraic hypersurface, as well as Haas'
combinatorial criterion for the maximality of plane curves near the tropical
limit.
