Solar geoengineering is the deliberate reduction in the absorption of incoming solar radiation by the Earth's climate system with the aim of reducing impacts of anthropogenic climate change. Climate model simulations project a diversity of regional outcomes that vary with the amount of solar geoengineering deployed. It is unlikely that a single small actor could implement and sustain global-scale geoengineering that harms much of the world without intervention from harmed world powers. However, a sufficiently powerful international coalition might be able to deploy solar geoengineering. Here, we show that regional differences in climate outcomes create strategic incentives to form coalitions that are as small as possible, while still powerful enough to deploy solar geoengineering. The characteristics of coalitions to geoengineer climate are modeled using a 'global thermostat setting game' based on climate model results. Coalition members have incentives to exclude non-members that would prevent implementation of solar geoengineering at a level that is optimal for the existing coalition. These incentives differ markedly from those that dominate international politics of greenhouse-gas emissions reduction, where the central challenge is to compel free riders to participate.