Assume $V=L(\mathbb{R})$ and the Axiom of Determinacy. Is every set of reals generated by ordinal-definable sets of reals under the operations of countable union and intersection?

The class of sets generated in this way is Wadge-cofinal and not wellorderable (it contains $\lbrace x\rbrace$ for every $x \in \mathbb{R}$) so there don't seem to be obvious limitations on its extent.

This question came up when I was trying to answer Asaf Karagila's "bonus question" here: http://mathoverflow.net/questions/102000/generating-family-for-the-lebesgue-sigma-algebra

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# sigma-algebra generated by OD sets

Assume $V=L(\mathbb{R})$ and the Axiom of Determinacy. Is every set of reals generated by ordinal-definable sets of reals under the operations of countable union and intersection?

This came up when I was trying to answer Asaf Karagila's "bonus question" here: http://mathoverflow.net/questions/102000/generating-family-for-the-lebesgue-sigma-algebra