Short gamma-ray bursts are thought to result from the mergers of two neutron stars or a neutron star and stellar mass black hole. The final stages of the merger are generally accompanied by the production of one or more tidal tails of ejecta, which fall back onto the remnant-disc system at late times. Using the results of a linear stability analysis, we show that if the material comprising these tails is modeled as adiabatic and the effective adiabatic index satisfies $gamma ge 5/3$, then the tails are gravitationally unstable and collapse to form small-scale knots. We analytically estimate the properties of these knots, including their spacing along the tidal tail and the total number produced, and their effect on the mass return rate to the merger remnant. We perform hydrodynamical simulations of the disruption of a polytropic (with the polytropic and adiabatic indices $gamma$ equal), $gamma =2$ neutron star by a black hole, and find agreement between the predictions of the linear stability analysis and the distribution of knots that collapse out of the instability. The return of these knots to the black hole induces variability in the fallback rate, which can manifest as variability in the lightcurve of the GRB and -- depending on how rapidly the instability operates -- the prompt emission. The late-time variability induced by the return of these knots is also consistent with the extended emission observed in some GRBs.
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