$\omega_1$ is inaccessible to the reals if and only if for all $x \in {}^\omega\omega$, $\omega_1^{L[x]} < \omega_1$.

The question is if $\omega_1$ is inaccessible to the reals in $V$ and $\mathbb{P}$ is an $\aleph_1$-preserving forcing, in the generic $\mathbb{P}$-extension $V[G]$, is $\omega_1$ still inaccessible to the reals?

Being inaccessible to the reals is a $\mathbf{\Pi}_4^1$ statement. Under certain assumptions, any forcing will preserve $\omega_1$ is inaccessible to the reals.

Therefore, a more specific question would be if $\kappa$ is an inaccessible cardinal in $L$ and $G$ is generic for $\text{Coll}(\omega, <\kappa)$, is $\omega_1$ being inaccessible to the reals preserved in all $\aleph_1$-preserving forcings extension of $L[G]$?

In $L[G]$, the $\text{Coll}(\omega, \omega_1^{L[G]})$ extension of $L[G]$ is equal to a $\text{Coll}(\omega, \kappa)$ extension of $L$. So $L[G]$ is an example of a model where arbitrary forcings do not preserve $\omega_1$ being inaccessible to the reals.

Thanks for any information.