We study dense colloidal crystals under oscillatory shear using a confocal microscope. At large strains the crystals yield and the suspensions form shear bands. The pure harmonic response exhibited by the suspension rules out the applicability of nonlinear rheology models typically used to describe shear banding in other types of complex fluids. Instead, we show that a model based on the coexistence of linearly responding phases of the colloidal suspension accounts for the observed flows. These results highlight a new use of oscillatory measurements in distinguishing the contribution of linear and nonlinear local rheology to a globally nonlinear material response.