Rossby waves are a type of global-scale wave that develops in planetary atmospheres, driven by the planet’s rotation¹. They propagate westward owing to the Coriolis force, and their characterization enables more precise forecasting of weather on Earth^2,3. Despite the massive reservoir of rotational energy available in the Sun’s interior and decades of observational investigation, their solar analogue defies unambiguous identification^4,​5,​6. Here we analyse a combined set of images obtained by the Solar TErrestrial RElations Observatory (STEREO) and the Solar Dynamics Observatory (SDO) spacecraft between 2011 and 2013 in order to follow the evolution of small bright features, called brightpoints, which are tracers of rotationally driven large-scale convection⁷. We report the detection of persistent, global-scale bands of magnetized activity on the Sun that slowly meander westward in longitude and display Rossby-wave-like behaviour. These magnetized Rossby waves allow us to make direct connections between decadal-scale solar activity and that on much shorter timescales. Monitoring the properties of these waves, and the wavenumber of the disturbances that they generate, has the potential to yield a considerable improvement in forecast capability for solar activity and related space weather phenomena.