Sunshade geoengineering - the installation of reflective mirrors between the Earth and the Sun to reduce incoming solar radiation, has been proposed as a mitigative measure to counteract anthropogenic global warming. Although the popular conception is that geoengineering can re-establish a ‘natural’ pre-industrial climate, such a scheme would itself inevitably lead to climate change, due to the different temporal and spatial forcing of increased CO₂ compared to reduced solar radiation. We investigate the magnitude and nature of this climate change for the first time within a fully coupled General Circulation Model. We find significant cooling of the tropics, warming of high latitudes and related sea ice reduction, a reduction in intensity of the hydrological cycle, reduced ENSO variability, and an increase in Atlantic overturning. However, the changes are small relative to those associated with an unmitigated rise in CO₂ emissions. Other problems such as ocean acidification remain unsolved by sunshade geoengineering.