Context. The Milky Way (MW) and Andromeda (M 31) galaxies possess rotating planes of satellites. The formation of these planes has not been explained satisfactorily so far. It has been suggested that the MW and M 31 satellites are ancient tidal dwarf galaxies; this might explain their configuration. This suggestion gained support by an analytic backward-calculation of the relative MW–M 31 orbit in the MOND modified dynamics paradigm. The result implied that the galaxies experienced a close flyby 7–11 Gyr ago.Aims. Here we explore the Local Group history in MOND in more detail using a simplified first-ever self-consistent simulation. We describe the features induced by the encounter in the simulation and identify possible real counterparts of these features.Methods. The initial conditions were set to eventually roughly reproduce the observed MW and M 31 masses, effective radii, separation, relative velocity, and disk inclinations. We used the publicly available adaptive-mesh-refinement code Phantom of RAMSES.Results. Matter was transferred from the MW to M 31 along a tidal tail in the simulation. The encounter induced the formation of several structures resembling the peculiarities of the Local Group. Most notably are that 1) a rotating planar structure formed around M 31 from the transferred material. It had a size similar to the observed satellite plane and was oriented edge-on to the simulated MW, just as the real plane. 2) The same structure also resembled the tidal features observed around M 31 by its size and morphology. 3) A warp in the MW developed with an amplitude and orientation similar to that observed. 4) A cloud of particles formed around the simulated MW, with the extent of the actual MW satellite system. The encounter did not end by merging in a Hubble time. The simulated stellar disks also thickened as a result of the encounter.Conclusions. The simulation demonstrated that MOND might explain many peculiarities of the Local Group; this needs to be verified with additional simulations. The simulation moreover showed that tidal features observed in galaxies, usually interpreted as merger remnants, could have been formed by matter exchange during non-merging galactic flybys in some cases.Key words: local group / galaxies: interactions / galaxies: formation / galaxies: kinematics and dynamics / galaxies: structure / gravitation★ Three movies are available athttp://www.aanda.org