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Stone duality generalizes to an equivalence between the categories$\mathsf{Stone}^{\mathsf{R}}$ of Stone spaces and closed relations and$\mathsf{BA}^\mathsf{S}$ of boolean algebras and subordination relations.Splitting equivalences in $\mathsf{Stone}^{\mathsf{R}}$ yields a category thatis equivalent to the category $\mathsf{KHaus}^\mathsf{R}$ of compact Hausdorffspaces and closed relations. Similarly, splitting equivalences in$\mathsf{BA}^\mathsf{S}$ yields a category that is equivalent to the category$\mathsf{DeV^S}$ of de Vries algebras and compatible subordination relations.Applying the machinery of allegories then yields that$\mathsf{KHaus}^\mathsf{R}$ is equivalent to $\mathsf{DeV^S}$, thus resolving aproblem recently raised in the literature.  The equivalence between $\mathsf{KHaus}^\mathsf{R}$ and $\mathsf{DeV^S}$further restricts to an equivalence between the category ${\mathsf{KHaus}}$ ofcompact Hausdorff spaces and continuous functions and the wide subcategory$\mathsf{DeV^F}$ of $\mathsf{DeV^S}$ whose morphisms satisfy additionalconditions.This yields an alternative to de Vries duality. One advantage ofthis approach is that composition of morphisms is usual relation composition.

A generalization of de Vries duality to closed relations between compact Hausdorff spaces