I think the following result is related:
Theorem 1(Mack Stanley). Let $L_α$ be a minimal countable standard transitive model of ZFC. There exists a real $x_{nwg}$ having the following three properties:
(1) $x_{nwg} ∈ L_α$.
(2) $L_α[x_{nwg}] \models ZFC$.
(3) $x_{nwg}$ is not definably generic over any outer model of $L_α$ that does not already contain $x_{nwg}$.
Indeed Stanley proves something stronger. See his paper ``A Non-generic Real Incompatible with $0^{\#}$''.
But on the other hand, we have partial positive answers as well. For example see Stanley's paper ``Invisible genericity and $0^♯$''. What is proved in this paper is simply that some instances of any type of non-constructible object are class generic over $L$. See also the paper ''$^*$forcing'' by Garvin Melles.