Let

$V$ be a vector space and let

$G$ be a

*building set*, i.e. a finite collection of subspaces of the dual

${V}^{*}$, whose elements are indecomposable. The open set

$V-{\bigcup}_{{H}_{i}\in G}{H}_{i}^{\perp}$ has a natural emebdding in the product of the projective spaces

$\mathbb{P}(V/{H}_{i}^{\perp})$. The closure

${Y}_{G}$ of the image is the De Conciniâ€“Procesi model of

$G$. The variety

${Y}_{G}$, in the case where

$G$ is a braid arrangement, is connected with the real part of the closure of the moduli space

${\overline{M}}_{0,n}\left(\mathbb{R}\right)$ of marked rational curves. Starting with a combinatorial description of the homology of

$V-{\bigcup}_{{H}_{i}\in G}{H}_{i}^{\perp}$, it is possible to characterize the homology of

${\overline{M}}_{0,n}\left(\mathbb{R}\right)$. The author performs a similar analysis when

$G$ is a general building set. Using chains of blow down of real De Concini - Procesi models, the author obtains a description of the ring structure of the homology of

${Y}_{G}$. By using this method, the author also proves that the homology of

${\overline{M}}_{0,n}\left(\mathbb{R}\right)$ has no odd torsion.