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Abstract We continue the study of the theories of Baldwin–Shi hypergraphs from [5]. Restricting our attention to when the rank δ is rational valued, we show that each countable model of the theory of a given Baldwin–Shi hypergraph is isomorphic to a generic structure built from some suitable subclass of the original class used in the construction. We introduce a notion of dimension for a model and show that there is a an elementary chain $$\left\{ {\mathfrak{M}_\beta :\beta \leqslant \omega } \right\}$$ of countable models of the theory of a fixed Baldwin–Shi hypergraph with $$\mathfrak{M}_\beta \preccurlyeq \mathfrak{M}_\gamma $$ if and only if the dimension of $$\mathfrak{M}_\beta $$ is at most the dimension of $$\mathfrak{M}_\gamma $$ and that each countable model is isomorphic to some $$\mathfrak{M}_\beta $$ . We also study the regular types that appear in these theories and show that the dimension of a model is determined by a particular regular type. Further, drawing on a large body of work, we use these structures to give an example of a pseudofinite, ω -stable theory with a nonlocally modular regular type, answering a question of Pillay in [11].