Abstract
Quantum theory of geometrically frustrated systems is usually approached as a gauge theory where the local conservation law becomes the Gauss law. Here we show that it can do something fundamentally different: enforce a global conserved quantity via a non-perturbative tiling invariant, rigorously linking microscopic geometry to a new macroscopically phase-coherent state. In a frustrated bosonic model on the honeycomb lattice in the cluster-charging regime at fractional filling, this mechanism protects a conserved global quantum number, the sublattice polarization N = N_A - N_B. Quantum fluctuation drives the spontaneous symmetry breaking of this global U(1) symmetry to result in a supernematic (SN) phase -- an incompressible yet phase-coherent quantum state that breaks rotational symmetry without forming a superfluid or realizing topological order. This establishes a route to a novel quantum many-body state driven by combinatorial constraints.