A well known result (stated and credited to Todorcevic in "Semiselective Coideals", by Farah, Mathematika, 1997, but with antecedents going back to Mathias) says that, under the appropriate large cardinal hypothesis (enough to get all sets of reals in $L(\mathbb{R})$ to be universally Baire, say), a selective ultrafilter is $L(\mathbb{R})$-generic for $([\omega]^\omega,\subseteq^*)$.

It is also well-known that selective ultrafilters need not exist; Kunen showed that they are destroyed by iterating random forcing over a model of CH. More generally, Miller showed that $Q$-points are destroyed by iterating Laver (or Mathias) forcing, and Shelah produced a model without $P$-points.

Here's my (admittedly broad) question:

Let $\mathbb{P}$ be a nontrivial (say, infinite, separative) $\sigma$-closed notion of forcing which is in $L(\mathbb{R})$, by which I mean the underlying set, its elements, and its order are all in $L(\mathbb{R})$. Suppose that under CH one can define an ultrafilter $G$ in $\mathbb{P}$ which is generic over $L(\mathbb{R})$ (under suitable large cardinal hypothesis). Is there a general theorem which tells us that such a $G$ consistently does not exist?