Tropical anvil clouds play a large role in Earth’s radiation balance, but their effect on global warming is uncertain. The conventional paradigm for these clouds attributes their existence to the rapidly-declining convective mass flux below the tropopause, which implies a large source of detraining cloudy air there. Here, we test this paradigm by manipulating the sources and sinks of cloudy air in cloud-resolving simulations. We find that anvils form in our simulations because of the long lifetime of upper tropospheric cloud condensates, not because of an enhanced source of cloudy air below the tropopause. We further show that cloud lifetimes are long in the cold upper troposphere because the saturation specific humidity is much smaller there than the condensed water loading of cloudy updrafts, which causes evaporative cloud decay to act very slowly. Our results highlight the need for novel cloud-fraction schemes that align with this decay-centric framework for anvil clouds.