  Given a polyhedral complex C with convex support, we characterize, by a local
codimension-2 condition, polyhedral complexes that coarsen C. The proof of the
characterization draws upon a surprising general shortcut for showing that a
collection of polyhedra is a polyhedral complex and upon a property of
hyperplane arrangements which is equivalent, for Coxeter arrangements, to Tits'
solution to the Word Problem. The motivating special case, the case where C is
a complete fan, generalizes a result of Morton, Pachter, Shiu, Sturmfels, and
Wienand that equates convex rank tests with semigraphoids. The proof of the
main result also implies a special case of Tietze's convexity theorem. We also
prove oriented matroid versions of our results, obtaining, as a byproduct, an
oriented matroid version of Tietze's convexity theorem.
