QCD jets produced in heavy-ion collisions at LHC or RHIC energies partially evolve inside the produced hot and dense quark gluon plasma, offering unique opportunities to study QCD splitting processes in different backgrounds. Induced (modified) splittings are expected to be the most relevant mechanism driving the modifications of in-medium jets compared to vacuum jets for a wide sets of observables. Although color coherence among different emitters has been identified as an essential mechanism in studies of the QCD antenna radiation, it is usually neglected in the multi-gluon medium-induced cascade. This independent gluon emission approximation can be analytically proved to be valid in the limit of very large media, but corrections or modifications to it have not been computed before in the context of the evolution (or rate) equation describing the gluon cascade. We propose a modified evolution equation that includes corrections due to the interference of subsequent emitters. In order to do so, we first compute a modified splitting kernel following the usual procedure of factorizing it from the subsequent Brownian motion. The calculation is performed in the two-gluon configuration with no overlapping formation times, that is expected to provide the first correction to the completely independent picture.