  For a reductive group over an algebraically closed field of characteristic $p
> 0$ we construct the abelian category of perverse $\mathbb{F}_p$-sheaves on
the affine Grassmannian that are equivariant with respect to the action of the
positive loop group. We show this is a symmetric monoidal category, and then we
apply a Tannakian formalism to show this category is equivalent to the category
of representations of a certain affine monoid scheme. We also show that our
work provides a geometrization of the inverse of the mod $p$ Satake
isomorphism. Along the way we prove that affine Schubert varieties are globally
$F$-regular and we apply Frobenius splitting techniques to the theory of
perverse $\mathbb{F}_p$-sheaves.
