An exact matching prior is a prior such that an associated credible set is also a confidence set. We study the existence of matching priors for general families of credible regions. Our main result gives sufficient conditions for exact matching priors to exist on compact metric spaces, extending recent results for finite spaces. Using our results, we design two families of credible regions that admit matching priors that approximate posterior credible balls and highest-posterior-density regions, respectively, and show that they share some of their most important properties. Our proof of the main theorem uses tools from nonstandard analysis and establishes new results about the nonstandard extension of the Wasserstein metric which may be of independent interest.