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. $x_{nwg}\notin L_α$.
>
>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

- <cite authors="Stanley, M. C.">_Stanley, M. C._, [**A non-generic real incompatible with $0^\#$**](http://dx.doi.org/10.1016/S0168-0072(96)00033-4), Ann. Pure Appl. Logic 85, No. 2, 157-192 (1997). [ZBL0877.03025](https://zbmath.org/?q=an:0877.03025).</cite> (Also [on Stanley's homepage](http://www.math.sjsu.edu/~stanley/ngreal.pdf).)

But on the other hand, we have partial positive answers as well. For example see Stanley's paper 

- <cite authors="Stanley, M. C.">_Stanley, M. C._, [**Invisible genericity and $0^\#$**](http://dx.doi.org/10.2307/2586652), J. Symb. Log. 63, No. 4, 1297-1318 (1998). [ZBL0924.03097](https://zbmath.org/?q=an:0924.03097).</cite> (Also [on Stanley's homepage](http://www.math.sjsu.edu/~stanley/inv-gen-zerosharp.pdf).)

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](https://arxiv.org/abs/math/9209210)'' by Garvin Melles.