The class of Matlis domains is surprisingly broad. However, whether every domain of Krull dimension 1 is a Matlis domain does not appear to have resolved in the literature. We construct a class of examples of 1-dimensional domains (in fact, almost Dedekind domains) that are overrings of K[X,Y], but are not Matlis domains. These domains fit in a larger context of what we term "Pruefer sections" of Noetherian domains.