A model of the Axiom of Determinacy in which every set of reals is universally Baire

The consistency of the theory $\mathsf{Z}\mathsf{F}+{\mathsf{A}\mathsf{D}}_{\mathbb{R}}+$every set of reals is universally Baire'' is proved relative to $\mathsf{ZFC} + {}$there is a cardinal that is a limit of Woodin cardinals and of strong cardinals.‘’ The proof is based on the derived model construction, which was used by Woodin to show that the theory $\mathsf{Z}\mathsf{F}+{\mathsf{A}\mathsf{D}}_{\mathbb{R}}+$every set of reals is Suslin'' is consistent relative to $\mathsf{ZFC} + {}$there is a cardinal $\lambda$ that is a limit of Woodin cardinals and of $<\lambda$-strong cardinals.‘’ The ${\mathrm{\Sigma }}_1^2$ reflection property of our model is proved using genericity iterations as used by Neeman and Steel.