We perform a cosmological magnetohydrodynamic simulation of a Milky Way-like galaxy with $\gtrsim10^8$ star particles to study the formation of out-of-equilibrium stellar disc structures in a full cosmological setting. In the plane defined by the coordinate and velocity perpendicular to the mid-plane (vertical phase space, $\{Z,V_Z\}$), stars in Solar-like volumes at late times exhibit clear spirals similar in shape and amplitude of the Gaia "Snail shell" phase spiral. We show that the phase spiral forms at a look back time of $\sim 6$ Gyr during the pericentric passage of a $\sim10^{10}$ $\rm M_{\odot}$ satellite which stimulates the formation of a resonant wake in the dark matter halo. The magnitude of the wake-induced gravitational torque at the Solar radius at this time is $\sim 8$ times that from the satellite, and leads to the formation of a disc warp that wraps up into a vertical phase spiral over time. This link between dark matter wakes and the formation of the phase spiral is first explicitly established here, and contrasts with earlier studies favouring direct torques from a Sgr dwarf galaxy or buckled bar origin. Furthermore, the feature is ever-present during the epoch of disc evolution: the initial disc is never featureless and unperturbed as is ubiquitously assumed in non-cosmological models. Our results demonstrate the highly complex and substantial role of the dark halo and its population of satellites on the dynamical history of the Milky Way's disc.