MacNeille completions of subordination algebras

S5-subordination algebras are a natural generalization of de Vries algebras. Recently it was proved that the category SubS5 of S5-subordination algebras and compatible subordination relations between them is equivalent to the category of compact Hausdorff spaces and closed relations. We generalize MacNeille completions of boolean algebras to the setting of S5-subordination algebras, and utilize the relational nature of the morphisms in SubS5 to prove that the MacNeille completion functor establishes an equivalence between SubS5 and its full subcategory consisting of de Vries algebras. We also show that the round ideal functor establishes a dual equivalence between SubS5 and the category of compact regular frames and preframe homomorphisms. Our results are choice-free and provide further insight into Stone-like dualities for compact Hausdorff spaces with various morphisms between them. In particular, we show how they restrict to the wide subcategories of SubS5 corresponding to continuous relations and continuous functions between compact Hausdorff spaces.