Precipitation and atmospheric circulation are the coupled processes through which tropical ocean surface temperatures drive global weather and climate1-5. Local sea surface warming tends to increase precipitation, but this local control is difficult to disentangle from remote effects of conditions elsewhere. As an example of such a remote effect, El Niño Southern Oscillation (ENSO) events in the equatorial Pacific Ocean alter precipitation across the tropics. Atmospheric circulations associated with tropical precipitation are predominantly deep, extending up to the tropopause. Shallow atmospheric circulations6-8 affecting the lower troposphere also occur, but the importance of their interaction with precipitation is unclear. Uncertainty in precipitation observations9,10 and limited observations of shallow circulations11 further obstruct our understanding of the ocean's influence on weather and climate. Despite decades of research, persistent biases remain in many numerical model simulations12-18, including excessively wide tropical rainbands14,18, the 'double-intertropical convergence zone problem'12,16,17 and too-weak responses to ENSO15. These biases demonstrate gaps in our understanding, reducing confidence in forecasts and projections. Here we use observations to show that seasonal tropical precipitation has a high sensitivity to local sea surface temperature. Our best observational estimate is an 80 per cent change in precipitation for every gram per kilogram change in the saturation specific humidity (itself a function of the sea surface temperature). This observed sensitivity is higher than in 43 of the 47 climate models studied, and is associated with strong shallow circulations. Models with more realistic (closer to 80%) sensitivity have smaller biases across a wide range of metrics. Our results apply to both temporal and spatial variation, over regions where climatological precipitation is about one millimetre per day or more. Our analyses of multiple independent observations, physical constraints and model data underpin these findings. The spread in model behaviour is further linked to differences in shallow convection, thus providing a focus for accelerated research to improve seasonal forecasts through multidecadal climate projections.