The author performs a nonlinear stability analysis using a multi-scales perturbation procedure for the instability of two layers of immiscible, inviscid, arbitrarily compressible fluids in relative motion. For modes of all wavenumbers on, or in the stable neighbourhood of, the linear neutral curve, the nonlinear evolution of the amplitude of the linear fields on the slow first-order scales is shown to be governed by a complicated nonlinear Klein-Gordon equation. Both the spatially dependent and space-independent versions of this equation are considered to obtain the regimes of physical parameter space where the linearly unstable solutions either evolve to final permanent envelope wave patterns resembling the ensembles of interactive vortices observed empirically, or are disrupted via nonlinear modulation instability.