  Given the tropicalization of a complex subvariety of the torus, we define a
morphism between the tropical cohomology and the rational cohomology of their
respective tropical compactifications. We say that the subvariety of the torus
is cohomologically tropical if this map is an isomorphism for all closed strata
of the tropical compactification.
  We prove that a sch\"on subvariety of the torus is cohomologically tropical
if and only if it is wundersch\"on and its tropicalization is a tropical
homology manifold. The former property means that the open strata in the
boundary of a tropical compactification are all connected and the mixed Hodge
structures on their cohomology are pure of maximum possible weight; the latter
property requires that, locally, the tropicalization verifies tropical
Poincar\'e duality.
  We study other properties of cohomologically tropical and wundersch\"on
varieties, and show that in a semistable degeneration to an arrangement of
cohomologically tropical varieties, the Hodge numbers of the smooth fibers are
captured in the tropical cohomology of the tropicalization. This extends the
results of Itenberg, Katzarkov, Mikhalkin, and Zharkov.
