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We develop a formalism of unit $F$-modules in the style of Lyubeznik andEmerton-Kisin for rings which have finite $F$-representation type afterlocalization and completion at every prime ideal. As applications, we show thatif $R$ is such a ring then the iterated local cohomology modules $H^{n_1}_{I_1}\circ \cdots \circ H^{n_s}_{I_s}(R)$ have finitely many associated primes, andthat all local cohomology modules $H^n_I(R / gR)$ have closed support when $g$is a nonzerodivisor on $R$.

A formalism of $F$-modules for rings with complete local finite $F$-representation type