We study the connection between multimatroids and moduli spaces of rational curves with cyclic action. Multimatroids are generalizations of matroids and delta-matroids introduced by Bouchet, which naturally arise in topological graph theory. The vantage point of moduli of curves provides a tropical framework for studying multimatroids, generalizing the previous connection between type-A permutohedral varieties (Losev--Manin moduli spaces) and matroids, and the connection between type-B permutohedral varieties (Batyrev--Blume moduli spaces) and delta-matroids. Specifically, we equate a combinatorial nef cone of the moduli space with the space of $\mathbb{R}$-multimatroids, a slight generalization of multimatroids, and we introduce the independence polytopal complex of a multimatroid, whose volume is identified with an intersection number on the moduli space. As an application, for the generating set of the Chow ring of the moduli space consisting of all psi-classes and their pullbacks along certain forgetful maps, we give a combinatorial formula for their intersection numbers by relating to the volumes of independence polytopal complexes of multimatroids.