for a tracial $*$-probability space $(\mathcal{a}, \varphi)$, c\'{e}bron, dahlqvist, and male constructed an enveloping traffic space $(\mathcal{g}(\mathcal{a}), \tau_\varphi)$ that extends the trace $\varphi$. the cdm construction provides a universal object that allows one to appeal to the traffic probability framework in generic situations, prioritizing an understanding of its structure. we show that $(\mathcal{g}(\mathcal{a}), \tau_\varphi)$ comes equipped with a canonical free product structure, regardless of the choice of $*$-probability space $(\mathcal{a}, \varphi)$. if $(\mathcal{a}, \varphi)$ is itself a free product, then we show how this additional structure lifts into $(\mathcal{g}(\mathcal{a}), \tau_\varphi)$. here, we find a duality between classical independence and free independence. we apply our results to study the asymptotics of large (possibly dependent) random matrices, generalizing and providing a unifying framework for results of bryc, dembo, and jiang and of mingo and popa. this is joint work with camille male.