  We show that the statement "every universally Baire set of reals has the
perfect set property" is equiconsistent modulo ZFC with the existence of a
cardinal that we call a virtually Shelah cardinal. These cardinals resemble
Shelah cardinals but are much weaker: if $0^\sharp$ exists then every Silver
indiscernible is virtually Shelah in $L$. We also show that the statement
$\text{uB} = {\bf\Delta}^1_2$, where $\text{uB}$ is the pointclass of all
universally Baire sets of reals, is equiconsistent modulo ZFC with the
existence of a $\Sigma_2$-reflecting virtually Shelah cardinal.
