  We study some power operations for ordinary $C_2$-equivariant homology with
coefficients in the constant Mackey functor $\underline{\mathbb{F}}_2$. In
addition to a few foundational results, we calculate the action of these power
operations on a $C_2$-equivariant dual Steenrod algebra. As an application, we
give a cellular construction of the $C_2$ equivariant Brown-Peterson spectrum
$\text{BP}\mathbf{R}$ and deduce its slice tower.
