I will explain the construction of a new class of Liouville domains that live in a complex torus of arbitrary dimension, whose boundary dynamics encodes information about the singularities of a toric compactification. The primary motivation for this work is to find a symplectic interpretation of some curious Laurent polynomials that appear in mirror symmetry for Fano manifolds; it also potentially opens a path to bound symplectic capacities of polarized projective varieties from below.