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An algorithm for computing compatibly Frobenius split subvarieties

Let $R$ be a ring of prime characteristic $p$, and let $F^e_* R$ denote $R$ viewed as an $R$-module via the $e$th iterated Frobenius map. Given a surjective map $ϕ: F^e_* R \to R$ (for example a Frobenius splitting), we exhibit an algorithm which produces all the $ϕ$-compatible ideals. We also explore a variant of this algorithm under the hypothesis that $ϕ$ is not necessarily a Frobenius splitting (or even surjective). This algorithm, and the original, have been implemented in Macaulay2.