Failure of inaccessibility in L produces a real with correct omega-one

The ambient ${\omega }_1$ is regular by Countable Choice, and it is a cardinal in $L$: an $L$-definable surjection from a smaller ordinal onto ${\omega }_1$ would still be a surjection in the universe. It is also regular in $L$, since an $L$-cofinal map from a smaller ordinal would remain cofinal in the universe.

Hence, if ${\omega }_1$ is not inaccessible in $L$, it fails one of the three clauses of [F4] there. It is uncountable in $L$ (it is uncountable in the universe and $L$ has the same ordinals) and regular in $L$ by step 1.1, so it is not a strong limit cardinal of $L$: there is a cardinal $\mu <{\omega }_1$ of $L$ with $(2^{\mu }{)}^L\ge {\omega }_1$. By GCH in $L$, $(2^{\mu }{)}^L={\mu }^{+}$, so the ordinal ${\omega }_1$ equals $({\mu }^{+}{)}^L$ for some $L$-cardinal $\mu <{\omega }_1$. This $\mu$ is infinite, since the $L$-successor of a finite cardinal is finite whereas the ambient ${\omega }_1$ is uncountable.

The ordinal $\mu$ is countable in the ambient universe because $\mu <{\omega }_1$, so there exists a real $x$ coding a bijection $b:\omega \to \mu$. This is one existential choice from a nonempty set of codes and needs no family-choice principle.

Let $\beta <({\mu }^{+}{)}^L={\omega }_1$. If $\beta =0$, it is countable in $L[x]$ trivially. If $\beta >0$, then Choice in $L$ and the fact that $\mu$ is an infinite $L$-cardinal imply that $L$ contains a surjection from $\mu$ onto $\beta$; this map also belongs to $L[x]$ by [F3]. Composing it with the bijection $b$ coded by $x$ makes $\beta$ countable in $L[x]$. Thus every ordinal below the ambient ${\omega }_1$ is countable in $L[x]$, so ${\omega }_1^{L[x]}\ge {\omega }_1$. Conversely the ambient ${\omega }_1$ is uncountable in $L[x]$, since any bijection with $\omega$ in $L[x]\subseteq V$ would contradict its ambient definition; hence ${\omega }_1^{L[x]}\le {\omega }_1$. Therefore equality holds.

The steps above produce a real $x$ with ${\omega }_1^{L[x]}={\omega }_1$ from the failure of inaccessibility in $L$; only this existential real is claimed, not the statement for every real.