Interactions between galaxies are common, and influence physical properties such as the global morphology and star-formation rate (Hubble type). Galaxies can interact in many different ways: they can merge together; they can pass through each other, with gas being stripped from the smaller of the two and compressed in the larger; and they can interact gravitationally (including, for example, tides in clusters). The relative importance of these mechanisms is often not clear, as the strength of each depends on poorly known parameters such as the density, extent and nature of the dark-matter haloes that surround galaxies. A nearby example of a galaxy interaction where the mechanism is controversial is that between our Galaxy and two of its neighbours, the Magellanic Clouds. Here we present the results of an atomic-hydrogen survey that help to elucidate this mechanism. Our data reveal a new stream of gas that lies in the opposite direction to the trailing Magellanic Stream and leads the motion of the Clouds. The existence of both leading and trailing streams supports a gravitational interaction whereby the streams are torn from the bodies of the Magellanic Clouds by tidal forces.