As an amoeba is the set of lengths of points in a variety, its coamoeba is the set of angles (arguments) of its points. The set of limiting arguments forms its phase limit set. This combinatorial backbone of the coamoeba reflects the structure of the corresponding tropical variety. We give a recursive description of the phase limit set of a linear space/hyperplane complement in terms of the flats of the hyperplane arrangement. We use this to study the phase limit set of a reduced discriminant, showing that it is a union of prisms over discriminants of lower dimension. We conjecture that in dimension at least three the discriminant is a subset of its phase limit set, which implies that that coamoeba discriminant has a polyhedral structure.