The non-linear stability of plane Poiseuille–Couette flow to three-dimensional disturbances is investigated asymptotically at large values of the Reynolds number R based on channel half-width and the maximum velocity of the Poiseuille component. The asymptotic theory, aimed at a detailed understanding of the physical mechanisms governing the amplitude-dependent stability properties of the flow, shows that the phase shifts induced across the critical layer and a near-wall shear layer are comparable when the disturbance size Δ=O(R−4/9)⁠. In addition, it emerges that at this crucial size both streamwise and spanwise wavelengths of the travelling wave disturbance are comparable with the channel width, with an associated phasespeed of O(1)⁠. Neutral solutions are found to exist in the range 0<V<2 with c0=V to leading order, where c0 and V are non-dimensional quantities representing the dominant phasespeed of the non-linear travelling waves and the wall sliding speed respectively. Moreover, these instability modes exist at sliding speeds well in excess of the linear instability cut-off. The amplitude equation governing these modes is derived analytically and we further find that this asymptotic structure breaks down in the limit V→2 when the disturbance streamwise wavelength decreases to O(R−1/3) and the maximum of the basic flow becomes located at the upper wall.