Summary: We study the topology of tropical varieties that arise from a certain natural class of varieties. We use the theory of tropical degenerations to construct a natural, “multiplicity-free” parameterization of \(\text{Trop}(X)\) by a topological space \(\Gamma_X\) and give a geometric interpretation of the cohomology of \(\Gamma_X\) in terms of the action of a monodromy operator on the cohomology of \(X\). This gives bounds on the Betti numbers of \(\Gamma_X\) in terms of the Betti numbers of \(X\) which constrain the topology of \(\operatorname{Trop}(X)\). We also obtain a description of the top power of the monodromy operator acting on middle cohomology of \(X\) in terms of the volume pairing on \(\Gamma_X\).