For multiprimary displays that have four or more primaries, a color may be reproduced using multiple alternative control vectors. We provide a complete characterization of the Metameric Control Set (MCS), i.e., the set of control vectors that reproduce a given color on the display. Specifically, we show that MCS is a convex polytope whose vertices are control vectors obtained from (parallelepiped) tilings of the gamut, i.e., the range of colors that the display can produce. The mathematical framework that we develop: (a) characterizes gamut tilings in terms of fundamental building blocks called facet spans, (b) establishes that the vertices of the MCS are fully characterized by the tilings of the gamut, and (c) introduces a methodology for the efficient enumeration of gamut tilings. The framework reveals the fundamental inter-relations between the geometry of the MCS and the geometry of the gamut developed in a companion Part I paper, and provides insight into alternative strategies for color control. Our characterization of tilings and the strategy for their enumeration also advance knowledge in geometry, providing new approaches and computational results for the enumeration of tilings for a broad class of zonotopes in R³.