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.

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

• Stanley, M. C., A non-generic real incompatible with $0^\#$, Ann. Pure Appl. Logic 85, No. 2, 157-192 (1997). ZBL0877.03025. (Also on Stanley's homepage.)

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

• Stanley, M. C., Invisible genericity and $0^\#$, J. Symb. Log. 63, No. 4, 1297-1318 (1998). ZBL0924.03097. (Also on Stanley's homepage.)

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.

I think the following results are 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 $0^{\#}$''.

Theorem 2(Sy Friedman) There is a real $R < 0^{\sharp}$ such that $L$ and $L[R]$ have the same cofinalities and $R$ is not an element of $L[G]$ for any $G$ which is generic for a tame class forcing over a ground model of the form $(L, A)$.

See theorem 5.2 of his book ''Fine structure and class forcing''

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.

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.