Every compact 3-Sasakian 7-manifold M admits a canonical 2-parameter family of co-closed \({\text {G}}_2\) -structures \(\varphi _{a,b}\) for \(a,b > 0\) , as well as a foliation by \(\varphi _{a,b}\) -associative 3-folds whose leaf space X is a positive quaternion-Kähler 4-orbifold. We prove that associative 3-folds in \((M,\varphi _{a,b})\) that are ruled by a certain type of geodesic are in correspondence with pseudo-holomorphic curves in the almost-complex 8-manifold \(Z \times S^2\) , where Z is the twistor space of X equipped with its strict nearly-Kähler structure. As an application, we construct infinitely many topological types of non-trivial, compact associative 3-folds in the squashed 7-spheres \((S^7, \varphi _{a,b})\) and squashed exceptional Aloff-Wallach spaces \((N_{1,1}, \varphi _{a,b})\) . Topologically, our examples are circle bundles over a genus g surface, for any \(g \ge 0\) .
