Let $P$ be the Levy-collapse of the ordinals, so $P$ is a class forcing notion that makes every ordinal countable.

Note that since $P$ is weakly homogeneous, for any formula $\phi(\overline{a})$ with parameters from $\mathbf{V}$, $P$ decides $\phi(\hat{\overline{a}})$. Hence it makes sense to ask whether $(HC, \in)$ is an elementary substructure of $(\mathbf{V}[G], \in)$ for some or any $P$-generic extension $\mathbf{V}[G]$.

Consider the axiom schema $\Phi$ that asserts that $(HC, \in)$ is elementary in $(\mathbf{V}[G], \in)$.

Question: Is $\Phi$ consistent relative to large cardinals, and if so, what is its strength? (Or at least get an upper bound)

As an example, $\Phi$ implies that $\omega_1$ is inaccessible to the reals. To see this, fix a real number $x$; then $\mathbf{L}[x]$ satisfies the powerset axiom, so $HC \models `` \mathbf{L}[x] \mbox{ satisfies the powerset axiom}"$, so $\mathbf{L}_{\omega_1}[x]$ satisfies the powerset axiom.