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

The consistency strength of the perfect set property for universally Baire sets of reals