  A detailed understanding of the moduli spaces $X(k,n)$ of $n$ points in
projective $k-1$ space is essential to the investigation of generalized
biadjoint scalar amplitudes, as discovered by Cachazo, Early, Guevara and
Mizera (CEGM) in 2019. But in math, conventional wisdom says that it is
completely hopeless due to the arbitrarily high complexity of realization
spaces of oriented matroids. In this paper, we nonetheless find a path forward.
  We present a Lie-theoretic realization of color factors for color-dressed
generalized biadjoint scalar amplitudes, formulated in terms of certain tilings
of the real moduli space $X(k,n)$ and collections of logarithmic differential
forms, resolving an important open question from recent work by Cachazo, Early
and Zhang. The main idea is to replace the realization space decomposition of
$X(k,n)$ with a large class of overlapping tilings whose topologies are
individually relatively simple. So we obtain a collection of color-dressed
amplitudes, each of which satisfies $U(1)$ decoupling separately. The essential
complexity appears when they are all superposed.
