We consider a popular model for long-duration gamma-ray bursts, in which the progenitor star, a stripped helium core, is spun up by tidal interactions with a black hole companion in a compact binary. We perform population synthesis calculations to produce a representative sample of such binaries, and model the effect that the companion has on material that falls back on to the newly formed black hole. Taking the results of hydrodynamic models of black hole formation by fallback as our starting point, we show that the companion has two principal effects on the fallback process. First, a break forms in the accretion curve at around 104 s. Secondly, subsequent to the break, we expect to see a flare of total energy around 1050 erg. We show that the break and flare times are set largely by the semimajor axis of the binary at the time of explosion, and that this correlates negatively with the flare energy. Although comparison with observations is non-trivial, we show that our predicted break times are comparable to those found in the X-ray light curves of canonical long-duration gamma-ray bursts. Similarly, the flare properties that we predict are consistent with the late-time flares observed in a subsample of bursts.