In the deep tropics, extensive anvil clouds produce a peak in cloud cover below the tropopause. The dominant paradigm for cloud cover attributes this anvil peak to a layer of enhanced mass convergence in the clear-sky upper-troposphere, which is presumed to force frequent detrainment of convective anvils. However, cloud cover also depends on the lifetime of cloudy air after it detrains, which raises the possibility that anvil clouds may be the signature of slow cloud decay rather than enhanced detrainment. Here we measure the cloud decay timescale in cloud-resolving simulations, and find that cloudy updrafts that detrain in the upper troposphere take much longer to dissipate than their shallower counterparts. We show that cloud lifetimes are long in the upper troposphere because the saturation specific humidity becomes orders of magnitude smaller than the typical condensed water loading of cloudy updrafts. This causes evaporative cloud decay to act extremely slowly, thereby prolonging cloud lifetimes in the upper troposphere. As a consequence, extensive anvil clouds still occur in a convecting atmosphere that is forced to have no preferential clear-sky convergence layer. On the other hand, when cloud lifetimes are fixed at a characteristic lower-tropospheric value, extensive anvil clouds do not form. Our results support a revised understanding of tropical anvil clouds, which attributes their existence to the microphysics of slow cloud decay rather than a peak in clear-sky convergence.