  Let $\Sigma$ be a fan inside the lattice $\mathbb{Z}^n$, and
$\mathcal{E}:\mathbb{Z}^n \rightarrow \operatorname{Pic}{S}$ be a map of
abelian groups. We introduce the notion of a principal toric fibration
$\mathcal{X}_{\Sigma, \mathcal{E}}$ over the base scheme $S$, relativizing the
usual toric construction for $\Sigma$. We show that the category of
ind-coherent sheaves on such a fibration is equivalent to the global section of
the Kashiwara-Schapira stack twisted by a certain local system of categories
with stalk $\operatorname{Ind}\operatorname{Coh} S$. It is a simultaneous
generalization of the work of Harder-Katzarkov [HK19] and of Kuwagaki [Kuw20],
and should be seen as a family-version of the coherent-constructible
correspondence [FLTZ11].
