We present the discovery of stellar tidal tails around the Large and the Small Magellanic Clouds (LMC and SMC, respectively) in the Gaia DR1 data. In between the Clouds, their tidal arms are stretched towards each other to form an almost continuous stellar bridge. Our analysis relies on the exquisite quality of the Gaia's photometric catalogue to build detailed star-count maps of the Clouds. We demonstrate that the Gaia DR1 data can be used to detect variable stars across the whole sky, and, in particular, RR Lyrae stars in and around the LMC and the SMC. Additionally, we use a combination of Gaia and GALEX to follow the distribution of Young Main Sequence stars in the Magellanic System. Viewed by Gaia, the Clouds show unmistakable signs of interaction. Around the LMC, clumps of RR Lyrae are observable as far as ∼20°, in agreement with the most recent map of Mira-like stars reported in Deason et al. The SMC's outer stellar density contours show a characteristic S-shape, symptomatic of the onset of tidal stripping. Beyond several degrees from the centre of the dwarf, the Gaia RR Lyrae stars trace the Cloud's trailing arm, extending towards the LMC. This stellar tidal tail mapped with RR Lyrae is not aligned with the gaseous Magellanic Bridge, and is shifted by some ∼5° from the Young Main Sequence bridge. We use the offset between the bridges to put constraints on the density of the hot gaseous corona of the Milky Way.