Total positivity for matroid Schubert varieties

We define the totally nonnegative matroid Schubert variety ${\mathcal{Y}}_V$ of a linear subspace $V\subset {\mathbb{R}}^n$. We show that ${\mathcal{Y}}_V$ is a regular CW complex homeomorphic to a closed ball, with strata indexed by pairs of acyclic flats of the oriented matroid of $V$. This closely resembles the regularity theorem for totally nonnegative generalized flag varieties. As a corollary, we obtain a regular CW structure on the real matroid Schubert variety of $V$.