  We show the existence of quasi-cluster $\mathcal{A}$-structures and cluster
Poisson structures on moduli stacks of sheaves with singular support in the
alternating strand diagram of grid plabic graphs by studying the microlocal
parallel transport of sheaf quantizations of Lagrangian fillings of Legendrian
links. The construction is in terms of contact and symplectic topology, showing
that there exists an initial seed associated to a canonical relative Lagrangian
skeleton. In particular, mutable cluster $\mathcal{A}$-variables are
intrinsically characterized via the symplectic topology of Lagrangian fillings
in terms of dually $\mathbb{L}$-compressible cycles. New ingredients are
introduced throughout this work, including the initial weave associated to a
grid plabic graph, cluster mutation along a non-square face of a plabic graph,
the concept of the sugar-free hull, and the notion of microlocal merodromy.
Finally, a contact geometric realization of the DT-transformation is
constructed for shuffle graphs, proving cluster duality for the cluster
ensembles.
