  Associated to any divisor in the Chow ring of a simplicial tropical fan, we
construct 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. We 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. For the Bergman fan of any matroid with building set, we prove that
there exists an open family of such cones of divisors with nonempty interiors.
We view the theory of normal complexes developed in this paper as a polytopal
model underlying the combinatorial Hodge theory pioneered by Adiprasito, Huh,
and Katz.
