  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 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 of finite structures with the inherited notion of strong
substructure. We introduce a notion of dimension for a model and show that
there is a an elementary chain $\{\mathfrak{M}_{\beta}:\beta<\omega+1\}$ 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 the work of Brody and Laskowski, we use these
structures to give an example of a pseudofinite, $\omega$-stable theory with a
non-locally modular regular type, answering a question of Pillay in $[9]$.
