Wave and sea ice properties in the Arctic and Southern Oceans are linked by feedback mechanisms, therefore the understanding of wave propagation in these regions is essential to model this key component of the Earth climate system. The most striking effect of sea ice is the attenuation of waves at a rate proportional to their frequency. The nonlinear Schrödinger equation (NLS), a fundamental model for ocean waves, describes the full growth-decay cycles of unstable modes, also known as modulational instability (MI). Here, a dissipative NLS (d-NLS) with characteristic sea ice attenuation is used to model the evolution of unstable waves. The MI in sea ice is preserved, however, in its phase-shifted form. The frequency-dependent dissipation breaks the symmetry between the dominant left and right sideband. We anticipate that this work may motivate analogous studies and experiments in wave systems subject to frequency-dependent energy attenuation.