Stratocumulus clouds cover 20% of the low-latitude oceans and are especially prevalent in the subtropics. They cool the Earth by shading large portions of its surface from sunlight. However, as their dynamical scales are too small to be resolvable in global climate models, predictions of their response to greenhouse warming have remained uncertain. Here we report how stratocumulus decks respond to greenhouse warming in large-eddy simulations that explicitly resolve cloud dynamics in a representative subtropical region. In the simulations, stratocumulus decks become unstable and break up into scattered clouds when CO2 levels rise above 1,200ppm. In addition to the warming from rising CO2 levels, this instability triggers a surface warming of about 8K globally and 10K in the subtropics. Once the stratocumulus decks have broken up, they only re-form once CO2 concentrations drop substantially below the level at which the instability first occurred. Climate transitions that arise from this instability may have contributed importantly to hothouse climates and abrupt climate changes in the geological past. Such transitions to a much warmer climate may also occur in the future if CO2 levels continue to rise.