We investigate the Kelvin–Helmholtz instability occurring at the interface of a shear-flow configuration in 2D compressible magnetohydrodynamics (MHD). The linear growth and the subsequent nonlinear saturation of the instability are studied numerically. We consider an initial magnetic field aligned with the shear flow, and analyse the differences between cases where the initial field is unidirectional everywhere (uniform case) and those where the field changes sign at the interface (reversed case). We recover and extend known results for pure hydrodynamic and MHD cases, with a discussion of the dependence of the nonlinear saturation on the wavenumber, the sound Mach number and the Alfvénic Mach number for the MHD case. A reversed field acts to destabilize the linear phase of the Kelvin–Helmholtz instability compared with the pure hydrodynamic case, while a uniform field suppresses its growth. In resistive MHD, reconnection events almost instantly accelerate the build up of a global plasma circulation. They play an important role throughout the further nonlinear evolution as well, since the initial current sheet is amplified by the vortex flow and can become unstable to tearing instabilities, forming magnetic islands. As a result, the saturation behaviour and the overall evolution of the density and the magnetic field are markedly different for the uniform versus the reversed-field case.