  We introduce the notion of ungraded matrix factorization as a mirror of
non-orientable Lagrangian submanifolds. An ungraded matrix factorization of a
polynomial $W$, with coefficients in a field of characteristic 2, is a square
matrix $Q$ of polynomial entries satisfying $Q^2 = W \cdot \mathrm{Id}$. We
then show that non-orientable Lagrangians correspond to ungraded matrix
factorizations under the localized mirror functor and illustrate this
construction in a few instances. Our main example is the Lagrangian submanifold
$\mathbb{R}P^2 \subset \mathbb{C}P^2$ and its mirror ungraded matrix
factorization, which we construct and study. In particular, we prove a version
of Homological Mirror Symmetry in this setting.
