Omega Centauri (omega Cen) is the Milky Way's most massive globular cluster, and has long been suspected of being the remnant core of an accreted dwarf galaxy. If this scenario is correct, omega Cen should be tidally limited and tidal debris should be spread along its orbit. Here we use N-body simulations to show that the recently discovered 'Fimbulthul' structure is the long-sought-for tidal stream of omega Cen, extending up to 28 degrees from the cluster. Follow-up high-resolution spectroscopy of five stream stars shows that they are closely grouped in velocity, and have metallicities consistent with having originated in that cluster. Informed by our N-body simulations, we devise a selection filter that we apply to Gaia mission data to also uncover the stream in the highly contaminated and crowded field within 10 degrees of omega Cen. Further modelling of the stream may help to constrain the dynamical history of the dwarf galaxy progenitor of this disrupting system and guide future searches for its remnant stars in the Milky Way.