Enhancement of the Earth’s albedo through the injection of sulfate aerosols into the stratosphere has been proposed as an approach to offset some of the adverse effects of climate change. Here we analyse an ensemble of simulations of the twenty-first century climate designed to explore a strategic geoengineering approach. Specifically, stratospheric sulfur injections are imposed at 15° and 30° in both hemispheres with the aim to minimize the changes in surface temperature, both in the global mean and in its gradients between hemispheres and from equator to pole. The approach accomplishes these goals and reduces previously noted adverse impacts of solar radiation management, such as excessive cooling in the tropics and weakening rainfall over land. Nonetheless, hydrological responses over the North Atlantic Ocean lead to an acceleration of the Atlantic meridional overturning circulation and to continued warming of the deep and polar oceans, particularly in the vicinity of southern Greenland. These changes could cause continued, albeit slower, cryospheric melt and global sea level rise. Our simulations demonstrate the complexity of the coupled climate response to geoengineering and highlight the need for significant advances in our ability to simulate the coupled climate system and the continued refinement of geoengineering strategies as a prerequisite to their successful implementation.