  In the present paper, we construct a $\mathbb{Z}/p$-equivariant analog of the
$\mathbb{Z}/2$-equivariant spectrum $BP\mathbb{R}$ previously constructed by Hu
and Kriz. We prove that this spectrum has some of the properties conjectured by
Hill, Hopkins, and Ravenel. Our main construction method is an
$\mathbb{Z}/p$-equivariant analog of the Brown-Peterson tower of $BP$, based on
a previous description of the $\mathbb{Z}/p$-equivariant Steenrod algebra with
constant coefficients by the authors. We also describe several variants of our
construction and comparisons with other known equivariant spectra.
