The Local Group is often seen to be a quiescent environment without significant merger events. However, an ancient major merger may have occurred in the most massive galaxy as suggested by the M31 classical bulge and its halo haunted by numerous stellar streams. Numerical simulations have shown that tidal tails formed during gas-rich major mergers are long-lived and could be responsible for old stellar streams and likely induce the formation of tidal dwarf galaxies (TDGs). Using several hydrodynamical simulations we have investigated the most prominent tidal tail formed during the first passage, which is gas rich and contains old and metal-poor stars. We discovered several striking coincidences after comparing its location and motion to those of the Milky Way (MW) and of the Magellanic Clouds (MCs). First, the tidal tail is sweeping a relatively small volume in which the MW precisely lies. Because the geometry of the merger is somehow fixed by the anisotropic properties of the giant stream (GS), we evaluate the chance of the MW to be at such a rendezvous with this gigantic tidal tail to be 5 per cent. Secondly, the velocity of the tidal tail matches the Large Magellanic Cloud (LMC) proper motion, and reproduces quite well the geometrical and angular momentum properties of the MW dwarfs, that is, the so-called disc of satellites, also known as the vast polar structure (VPOS). Thirdly, the simulation of the tidal tail reveals one of the formed TDGs with the mass and location almost comparable to those of the LMC. Our present modelling is, however, too limited to study the detailed interaction of gas-rich TDGs with the potential of the MW, and a complementary study is required to test whether the dwarf intrinsic properties can be accounted for by our scenario. Nevertheless this study suggests a causal link between an expected event, an ancient, gas-rich major merger at the M31 location, and several enigmas in the Local Group, namely the GS in the M31 outskirts, the VPOS almost perpendicular to the MW disc, and the presence of the MCs, two Irr galaxies near the MW.