  Let M be an arbitrary Riemannian homogeneous space, and let Omega be a space
of tilings of M, with finite local complexity (relative to some symmetry group
Gamma) and closed in the natural topology. Then Omega is the inverse limit of a
sequence of compact finite-dimensional branched manifolds. The branched
manifolds are (finite) unions of cells, constructed from the tiles themselves
and the group Gamma. This result extends previous results of Anderson and
Putnam, of Ormes, Radin and Sadun, of Bellissard, Benedetti and Gambaudo, and
of G\"ahler. In particular, the construction in this paper is a natural
generalization of G\"ahler's.
