Fomin-Shapiro, Lusztig, Galashin-Karp-Lam and several others have studied the images of certain maps to totally nonnegative spaces whose fibers encode the nonnegative real relations amongst exponentiated Chevalley generators. This talk will focus on the fibers of these maps. We prove that the stratification on these fibers induced by the natural stratification of $\mathbb{R}_{\ge 0}^d$ 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 the interior dual block complex of a subword complex. We prove that these interior dual block complexes of subword complexes are contractible. I will describe several of the ingredients that go into this work, providing background and examples along the way. This is joint work with Jim Davis and Ezra Miller.