Bergman fans of matroids are the tropical equivalent of linear spaces. In the context of tropical homology, such fans can be shown to satisfy tropical Poincaré duality, i.e. a duality between tropical homology and cohomology. Thanks to work of Amini and Piquerez, satisfying tropical Poincaré duality is related to the combinatorial Hodge theory of the Matroid Chow ring of Adiprasito-Huh-Katz. In this talk, we will give some results on the problem of classifying which fans satisfy tropical Poincaré duality, and give some perspectives on recent work relating tropical Poincaré duality and the cohomology of subvarieties of tori.