Let $\mathrm{Mat}_{n\times m}(\mathbb{C})$ be the affine space of $n\times m$ complex matrices, and let $\mathcal{Z}_{n,m,r}$ (resp. $\mathcal{UZ}_{n,m,r}$) be the locus in $\mathrm{Mat}_{n\times m}(\mathbb{C})$ corresponding to rook placements with exactly (resp. at least) $r$ rooks. The orbit harmonics method yields two quotient rings $R(\mathcal{Z}_{n,m,r})$ and $R(\mathcal{UZ}_{n,m,r})$, where both rings have the additional structures of $\mathfrak{S}_n\times\mathfrak{S}_m$-modules. We find the generators of their defining ideals and compute their graded Frobenius image. Furthermore, we give a nontrivial generalization of Viennot's shadow line avatar of the Schensted correspondence to rook placements in $\mathcal{UZ}_{n,m,r}$. This generalization is used to determine the standard monomial basis of $R(\mathcal{UZ}_{n,m,r})$ with respect to a diagonal term order. Joint with Jasper (Moxuan) Liu.