A statically stable stratified water column that also contains horizontal property contrasts (either of passive tracer alone or of two dynamically active solutes) is generated and continuously maintained for a long period by releasing two turbulent buoyant plumes of equal buoyancy fluxes into opposite ends of a long channel of water. The bottom outflows from the plumes also continuously excite internal gravity waves that produce a series of counter-flowing quasi-horizontal shear layers which are quasi-stationary relative to the box but whose phase propagates downward through the upward-moving water column. We report that the flow further involves an oscillation associated with the internal waves that gives rise to a sequence of interleaving intrusions across the horizontal gradient region. The wave-driven intrusions are advected upward with the ‘filling-box’ circulation and have the appearance of a spatially growing instability. The intrusions are examined in cases having no horizontal property differences other than a passive tracer. In further experiments where one plume is salt solution and the other is sugar solution, there is vigorous double-diffusive convection on the interleaving intrusions, including salt fingering and diffusive density interfaces, but this convection has only a weak influence on the intrusion thicknesses and velocities. We conclude that under all conditions attained in these experiments, the interleaving is driven by internal waves and not by the property gradients, and we infer that the wave-generated intrusions enhance double-diffusive buoyancy fluxes.