Context. The disc instability model (DIM) accounts well for most of the observed properties of dwarf novae and soft X-ray transients, but the rebrightenings, reflares, and echoes occurring at the end of outbursts or shortly after in WZ Sge stars or soft X-ray transients have not yet been convincingly explained by any model.Aims. We determine the additional ingredients that must be added to the DIM to account for the observed rebrightenings.Methods. We analyse in detail a recently discovered system, TCP J21040470+4631129, which has shown very peculiar rebrightenings. We also model the light curve of this system using our numerical code, including mass transfer variations from the secondary, inner–disc truncation, disc irradiation by a hot white dwarf and, in some cases, the mass-transfer stream over(under) flow.Results. We show that the luminosity in quiescence is dominated by a hot white dwarf that cools down on timescales of months. For a reason that remains elusive, the mass transfer rate from the secondary has to increase by several orders of magnitudes during the initial superoutburst. The mass transfer rate slowly returns to its secular average and causes the observed succession of outbursts with increasing quiescence durations until the disc can be steady, cold, and neutral; its inner parts are truncated either by the white dwarf magnetic field or by evaporation. The very short, quiescence phases between reflares are reproduced when the mass-transfer stream overflows the disc. Using similar additions to the DIM, we also produced light curves close to those observed in two WZ Sge stars, the prototype and EG Cnc.Conclusions. Our model successfully explains the reflares observed in WZ Sge systems. It requires, however, the inner disc truncation in dwarf novae to be due not only to the white dwarf magnetic field but, as in X-ray binaries, rather to evaporation of the inner disc. A similar model could also explain reflares observed in soft X-ray transients.Key words: accretion / accretion disks / stars: dwarf novae / instabilities