  We construct an exact completion for regular categories enriched in the
cartesian closed category $\mathsf{Pos}$ of partially ordered sets and monotone
functions by employing a suitable calculus of relations. We then characterize
the embedding of any regular category into its completion and use this to
obtain examples of concrete categories which arise as such completions. In
particular, we prove that the exact completion in this enriched sense of both
the categories of Stone and Priestley spaces is the category of compact ordered
spaces of L. Nachbin. Finally, we consider the relationship between the
enriched exact completion and categories of internal posets in ordinary
categories.
