Continued anthropogenic CO2 emissions are expected to impact tropical coral reefs by further raising sea surface temperatures (SST) and intensifying ocean acidification (OA). Although geoengineering by means of solar radiation management (SRM) may mitigate temperature increases, OA will persist, raising important questions regarding the impact of different stressor combinations. We apply statistical Bioclimatic Envelope Models to project changes in shallow water tropical coral reef habitat as a single niche (without resolving biodiversity or community composition) under various representative concentration pathway and SRM scenarios, until 2070. We predict substantial reductions in habitat suitability centered on the Indo-Pacific Warm Pool under net anthropogenic radiative forcing of ≥3.0 W/m2. The near-term dominant risk to coral reefs is increasing SSTs; below 3 W/m2 reasonably favorable conditions are maintained, even when achieved by SRM with persisting OA. “Optimal” mitigation occurs at 1.5 W/m2 because tropical SSTs overcool in a fully geoengineered (i.e., preindustrial global mean temperature) world.