Is it consistent that there is a model of $\mathsf{ZFC}$ (or $\mathsf{ZF}$) with the following properties:
(1) For all $x \in {}^\omega 2$, $x^\sharp$ exists (or $\mathbf{\Sigma}_1^1$ determinacy)
(2) For all sets $A$, there exists $x \in {}^\omega 2$ such that $A \in L[x]$, i.e. every set is constructible from a real.
If one takes an arbitrary model $V$ of $\mathsf{ZFC}$ satisfying (1), will $L[({}^\omega2)^V]$ have this property or does it necessarily add a set that is not constructible from any single real?
Thanks for any information or clarification.