We attempt to understand the planes of satellite galaxies orbiting the Milky Way (MW) and M31 in the context of Modified Newtonian Dynamics, which implies a close MW-M31 flyby occurred ≃8 Gyr ago. Using the timing argument, we obtain MW-M31 trajectories consistent with cosmological initial conditions and present observations. We adjust the present M31 proper motion within its uncertainty in order to simulate a range of orbital geometries and closest approach distances. Treating the MW and M31 as point masses, we follow the trajectories of surrounding test particle discs, thereby mapping out the tidal debris distribution. Around each galaxy, the resulting tidal debris tends to cluster around a particular orbital pole. We find some models in which these preferred spin vectors align fairly well with those of the corresponding observed satellite planes. The radial distributions of material in the simulated satellite planes are similar to what we observe. Around the MW, our best-fitting model yields a significant fraction (0.22) of counter-rotating material, perhaps explaining why Sculptor counter-rotates within the MW satellite plane. In contrast, our model yields no counter-rotating material around M31. This is testable with proper motions of M31 satellites. In our best model, the MW disc is thickened by the flyby 7.65 Gyr ago to a root mean square height of 0.75 kpc. This is similar to the observed age and thickness of the Galactic thick disc. Thus, the MW thick disc may have formed together with the MW and M31 satellite planes during a past MW-M31 flyby.