We explicitly describe the tropicalization of a cluster variety of finite type C, realizing it as the space of axially symmetric phylogenetic trees. We also find all occurring sign patterns of coordinates, for both the cluster variety and the cluster configuration space. We show that each of the corresponding signed tropicalizations is, combinatorially, dual to either a cyclohedron or an associahedron. As additional results, we construct Gröbner and tropical bases for the defining ideals of both varieties, and classify the arising toric degenerations.