Our story is motivated by the configuration space of particles on spheres. In the 1970s, Grothendieck, Deligne, and Mumford constructed a way to keep track of particle collisions in this space using Geometric Invariant Theory. In the 1990s, Gromov and Witten utilized them as invariants arising from string field theory and quantum cohomology. We consider the real points of these spaces, but now interpret them as spaces of rooted metric labeled trees. They have elegant geometric and combinatorial properties, being compact hyperbolic manifolds with a beautiful tessellation by convex polytopes. In recent years, they have gained importance in their own right, appearing in areas such as representation theory, geometric group theory, tropical geometry, and lately reinterpreted by Levy and Pachter as spaces of phylogenetic networks. In particular, these real moduli spaces resolve the singularities of the spaces of phylogenetic trees studied by Billera, Holmes, and Vogtmann.