We introduce a set of coupled equations for multi-layer water waves that removes the ill-posedness of the multi-layer Green–Naghdi (MGN) equations in the presence of shear. The new well-posed equations are Hamiltonian and in the absence of imposed background shear, they retain the same travelling wave solutions as MGN. We call the new model the square root depth (Inline Formula) equations from the modified form of their kinetic energy of vertical motion. Our numerical results show how the Inline Formula equations model the effects of multi-layer wave propagation and interaction, with and without shear.