We present a detailed analysis of two fields located 26 kpc (∼5 radial scalelengths) from the centre of M31 along the south-west semimajor axis of the disc. One field samples the major axis populations – the Outer Disc field – while the other is offset by ∼18 arcmin and samples the warp in the stellar disc – the warp field. The colour–magnitude diagrams (CMDs) based on Hubble Space Telescope Advanced Camera for Surveys imaging reach old main-sequence turn-offs (∼12.5 Gyr). We apply the CMD-fitting technique to the warp field to reconstruct the star formation history (SFH). We find that after undergoing roughly constant star formation until about 4.5 Gyr ago, there was a rapid decline in activity and then a ∼1.5 Gyr lull, followed by a strong burst lasting 1.5 Gyr and responsible for 25 per cent of the total stellar mass in this field. This burst appears to be accompanied by a decline in global metallicity which could be a signature of the inflow of metal-poor gas. The onset of the burst (∼3 Gyr ago) corresponds to the last close passage of M31 and M33 as predicted by detailed N-body modelling, and may have been triggered by this event. We reprocess the deep M33 outer disc field data of Barker et al. in order to compare consistently derived SFHs. This reveals a similar duration burst that is exactly coeval with that seen in the M31 warp field, lending further support to the interaction hypothesis. We reliably trace star formation as far back as 12–13 Gyr ago in the outer disc of M31, while the onset of star formation occurred about 2 Gyr later in M33, with median stellar ages of 7.5 and 4.5 Gyr, respectively. The complex SFHs derived, as well as the smoothly varying age–metallicity relations, suggest that the stellar populations observed in the far outer discs of both galaxies have largely formed in situ rather than migrated from smaller galactocentric radii. The strong differential reddening affecting the CMD of the Outer Disc field prevents derivation of the SFH using the same method. Instead, we quantify this reddening and find that the fine-scale distribution of dust precisely follows that of the H i gas. This indicates that the outer H i disc of M31 contains a substantial amount of dust and therefore suggests significant metal enrichment in these parts, consistent with inferences from our CMD analysis.