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We continue the study of the theories of Baldwin-Shi hypergraphs from $[5]$.Restricting our attention to when the rank $\delta$ is rational valued, we showthat each countable model of the theory of a given Baldwin-Shi hypergraph isisomorphic to a generic structure built from some suitable subclass of theoriginal class of finite structures with the inherited notion of strongsubstructure. We introduce a notion of dimension for a model and show thatthere is a an elementary chain $\{\mathfrak{M}_{\beta}:\beta<\omega+1\}$ ofcountable models of the theory of a fixed Baldwin-Shi hypergraph with$\mathfrak{M}_{\beta}\preccurlyeq\mathfrak{M}_\gamma$ if and only if thedimension 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 thesetheories and show that the dimension of a model is determined by a particularregular type. Further, drawing on the work of Brody and Laskowski, we use thesestructures to give an example of a pseudofinite, $\omega$-stable theory with anon-locally modular regular type, answering a question of Pillay in $[9]$.

Countable models of the theories of Baldwin-Shi hypergraphs and their regular types