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The reduction in global-mean precipitation when stratospheric aerosol geoengineering is used to counterbalance global warming from increasing carbon dioxide (CO₂) concentrations has been mainly attributed to the temperature-independent effect of CO₂ on atmospheric radiative cooling. We demonstrate here that stratospheric sulphate aerosol itself also acts to reduce global-mean precipitation independent of its effects on temperature. The temperature-independent effect of stratospheric aerosol geoenginering on global-mean precipitation is calculated by removing temperature-dependent effects from climate model simulations of the Geoengineering Model Intercomparison Project (GeoMIP). When sulphate aerosol is injected into the stratosphere at a rate of 5 Tg SO₂ per year the aerosol reduces global-mean precipitation by approximately 0.2 %, though multiple ensemble members are required to separate this effect from internal variability. For comparison, the precipitation reduction from the temperature-independent effect of increasing CO₂ concentrations under the RCP4.5 scenario of the future is approximately 0.5 %. The temperature-independent effect of stratospheric sulphate aerosol arises from the aerosol's effect on tropospheric radiative cooling. Radiative transfer calculations show this is mainly due to increasing downward emission of infrared radiation by the aerosol, but there is also a contribution from the stratospheric warming the aerosol causes. Our results suggest climate model simulations of solar dimming can capture the main features of the global-mean precipitation response to stratospheric aerosol geoengineering.