Tropical anvil clouds play a large role in Earth's radiation balance, but their effect on anthropogenic global warming is uncertain. Here, we use cloud-resolving simulations of tropical convection to invalidate the dominant paradigm for anvil clouds and to develop a replacement. The old paradigm attributes anvil clouds to the rapidly-declining convective mass flux below the tropopause, which implies a large source of detraining cloudy air there. However, by quantifying and manipulating the sources and sinks of cloudy air in the simulations, we demonstrate that anvil clouds actually form because of the long lifetime of cloud condensate in the upper troposphere, 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 typical condensed water loading of cloudy updrafts, causing evaporative cloud dissipation to act very slowly. Since the majority of contemporary global climate models do not include vertically-varying sinks of condensate in their cloud-fraction parameterizations, our results call for the development of novel schemes that align with the new decay-centric paradigm for anvil clouds. A parameterization based on cloud lifetimes might be trusted to predict changes in cloud fraction with global warming, and thereby determine whether anvil clouds produce a positive or negative radiative feedback.