Here we perform a series of high-resolution large eddy simulations capable of explicitly resolving stratocumulus clouds at scales of tens of meters to understand the details of their dynamical response to greenhouse warming. The LES are coupled to a slab ocean model to ensure a physically realizable surface energy budget and simulations are run until sea surface temperatures reach an approximate steady state. An innovative aspect of these simulations is the coupling between the state of the representative subtropical domain simulated by the LES and the large-scale forcings of heat and moisture applied to it, which are obtained through an idealized but physically relevant model of the tropical climate response. Using this framework, we identify a critical CO2 concentration above which stratocumulus cloud decks cannot be sustained. Subsequently, CO2 concentrations must be decreased significantly below this critical level to reform stratocumuli, demonstrating the hysteresis of the response to CO2.