When double neutron star or neutron star-black hole binaries merge, the final remnant may comprise a central solar-mass black hole surrounded by a 0.01-0.1 solar masses torus. The subsequent evolution of this disc may be responsible for short gamma-ray bursts (SGRBs). A comparable amount of mass is ejected into eccentric orbits and will eventually fall back to the merger site after approximately 0.01 seconds. In this Paper, we investigate analytically the fate of the fallback matter, which may provide a luminous signal long after the disc is exhausted. We find that matter in the eccentric tail returns at a super-Eddington rate and is eventually (0.1 sec) unable to cool via neutrino emission and accrete all the way to the black hole. Therefore, contrary to previous claims, our analysis suggests that fallback matter is not an efficient source of late time accretion power and is unlikely to cause the late flaring activity observed in SGRB afterglows. The fallback matter rather forms a radiation-driven wind or a bound atmosphere. In both cases, the emitting plasma is very opaque and photons are released with a degraded energy in the X-ray band. We therefore suggest that compact binary mergers could be followed by an X-ray renaissance, as late as several days to weeks after the merger. This might be observed by the next generation of X-ray detectors.
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