P. Di Francesco - University of Illinois and IPhT Saclay Alternating Sign Matrices (ASM) are at the confluent of many interesting combinatorial/algebraic problems: Laurent phenomenon for the octahedron equation, configurations of the Square Ice (Six Vertex model), Descending Plane Partitions (DPP), etc. Here we consider the Triangular Lattice version of the Ice model with suitable boundary conditions leading to an integrable 20 Vertex model. Configurations give rise to generalizations of ASM, which we coin Alternating Phase Matrices (APM). We generalize the ASM-DPP correspondence by showing that APM are equinumerous to the quarter-turn symmetric domino tilings of a quasi-Aztec square with a central cross-shaped hole, and obtain a compact determinant formula for their enumeration. We also present conjectures for triangular Ice with other types of boundary conditions, and results on the limit shape of large APM. (joint work with E. Guitter, IPhT Saclay, France).