Fomin-Shapiro, Lusztig, and several others have studied maps to totally nonnegative spaces whose fibers encode the nonnegative real relations amongst exponentiated Chevalley generators.   We prove that the stratification on each such fiber induced by the natural stratification of $\RR_{\ge 0}^d$ is a cell decomposition, doing so by providing a parametrization for each stratum.  We also show that the face poset for this cell decomposition is the face poset of a regular CW complex, namely of the interior dual block complex of a subword complex.  This talk will focus on examples illustrating some of the main ideas that go into this work as well as background, motivations, and some further open questions.  This is joint work with Jim Davis and Ezra Miller.