Exact coherent states in plane Couette flow are extended to an oblique parallelogram computational domain and the large-Reynolds-number asymptotic descriptions of the states are derived. When the tilt angle of the domain is increased, the states first obey vortex–wave interaction theory and then develop a new asymptotic structure. The latter asymptotic structure is characterized by a self-similar nonlinear interaction localized in the fluid layer. For the largest scale of the self-similar nonlinear structure, the theory predicts an inverse Reynolds number dependence for the tilt angle of the oblique pattern. That result is consistent with the numerical and experimental observations of the laminar–turbulent banded pattern in shear flows.