‘Phase-change’ memory materials, such as the canonical composition Ge2Sb2Te5, are being actively researched for non-volatile resistive random-access memory applications. In these devices, ultra-rapid reversible transformations between metastable highly electrically conducting (degenerate-semiconducting) crystalline and more electrically resistive (semiconducting) glassy phases are produced by the application of appropriate voltage pulses. Multilevel programming, wherein more than two metastable resistance states can be stored in the memory material as different proportions of partially glassy/crystalline regions, allows more than one bit to be stored per memory cell. However, this route to increasing data density, without recourse to device-size down-scaling, is threatened by the phenomenon of ‘resistance drift’, wherein the electrical resistance of the glassy phase slowly increases with time, following a weak power-law dependence, after being written with a voltage pulse. In this paper, we propose an intrinsic electronic mechanism for the resistance drift by identifying it with the phenomenon of persistent photoconductivity that is commonly observed in a wide range of disordered semiconductors. We develop a model for it in terms of the long-time, deep-trap release and subsequent recombination of charge carriers, akin to that which is believed to be responsible for the long-time photocurrent decay in amorphous semiconductors, such as hydrogenated amorphous silicon. In this case, the parameters controlling the resistance drift are the widths of the (localized) valence- and conduction-band tails in the vicinity of the bandgap. Hence, there is the potential for mitigating resistance drift in the amorphous state of phase-change memory materials by suitable material engineering (e.g. via compositional or fabricational control) to control the extent of band-tailing, thereby facilitating the future introduction of multistate memory.