With the Barwise compactness theorem in mind, say that a countable infinite ordinal $\alpha$ is $\omega_1$-Barwise iff whenever $T$ is a $\Sigma_1(L_\alpha)$ set of $\mathcal{L}_{\infty,\color{red}{\omega_1}}$-sentences and every subtheory of $T$ which is an element of $L_\alpha$ is satisfiable, then $T$ itself is satisfiable. Such ordinals do in fact exist by an elementary argument which in fact applies to pretty much any "$\mathsf{HC}$-coded" logic.
I'm broadly curious about the $\omega_1$-Barwise ordinals. At the above-linked question I asked how large they must be; here I'd like to ask a somewhat different question, namely whether $\omega_1$-Barwiseness can be approximated in a finitary first-order way:
Is there a first-order theory $\mathsf{B}$ in the language of set theory such that $(i)$ uncountably many countable levels of $L$ satisfy $\mathsf{B}$ and $(ii)$ whenever $\alpha$ is countable and $L_\alpha\models \mathsf{B}$ we have that $\alpha$ is $\omega_1$-Barwise?
Recall that Barwise showed that exactly this happens at $\mathcal{L}_{\infty,\omega}$: if $L_\alpha\models\mathsf{KP}$ then we get the usual Barwise compactness property at $L_\alpha$. So this isn't completely ludicrous (although I am highly confident that the answer is negative).