  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 $V$.

Combinatorics;Algebraic Geometry