The phase rank of a matrix

In this paper, we introduce and study the notion of phase rank. Given a matrix filled with phases, i.e., with complex entries of modulus $1$, its phase rank is the smallest possible rank for a complex matrix with the same phase, but possible different modulus. This notion generalizes the notion of sign rank, and is the complementary notion to that of phaseless rank. It also has intimate connections to the problem of ray nonsingularity and provides an important class of examples for the study of coamoebas. Among other results, we provide a new characterization of phase rank $2$ for $3 \times 3$ matrices, that implies the new result that for the $3\times 3$ determinantal variety, its coamoeba is determined solely by the colopsidedness criterion.