One of the first large cloud systems ever observed on Titan was a stationary event at the southern pole that lasted almost two full Titan days. Its stationary nature and large extent are puzzling given that low-level winds should transport clouds eastward, pointing to a mechanism such as atmospheric waves propagating against the mean flow. We use a composite of 47 large convective events across 15 Titan years of simulations from the Titan Atmospheric Model to show that Rossby waves trigger polar convection-which halts the waves and produces stationary precipitation-and then communicate its impact globally. In the aftermath of the convection, forced waves undergo a complicated evolution, including cross-equatorial propagation and tropical-extratropical interaction. The resulting global impact from convection implies its detectability anywhere on Titan, both via surface measurements of pressure and temperature and through remote observation of the outgoing longwave radiation, which increases by ~0.5% globally.