  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 that
is equivalent to the category $\mathsf{KHaus}^\mathsf{R}$ of compact Hausdorff
spaces 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 a
problem recently raised in the literature.
  The equivalence between $\mathsf{KHaus}^\mathsf{R}$ and $\mathsf{DeV^S}$
further restricts to an equivalence between the category ${\mathsf{KHaus}}$ of
compact Hausdorff spaces and continuous functions and the wide subcategory
$\mathsf{DeV^F}$ of $\mathsf{DeV^S}$ whose morphisms satisfy additional
conditions.This yields an alternative to de Vries duality. One advantage of
this approach is that composition of morphisms is usual relation composition.
