Associated to any divisor in the Chow ring of a simplicial tropical fan, we discuss the construction a family of polytopal complexes, called normal complexes, which we propose as an analogue of the well-studied notion of normal polytopes from the setting of complete fans. The talk will describe certain closed convex polyhedral cones of divisors for which the “volume” of each divisor in the cone—that is, the degree of its top power—is equal to the volume of the associated normal complexes. We will discuss the theory of normal complexes developed in talk as a polytopal model underlying the combinatorial Hodge theory pioneered by Adiprasito, Huh, and Katz.