What is forcing indescribability? Suppose $m, n\in\omega$ and $\kappa$ is a cardinal. Then $\kappa$ is $\Pi^m_n$-indescribable if every $\Pi^m_n$-sentence true about $\kappa$ is true about some $\lambda<\kappa$; formally, if for every $\Pi_n$-sentence $\varphi$ in the language of set theory with a unary predicate and every $A\subseteq V_\kappa$, there is some $\lambda<\kappa$ such that $$(V_{\kappa+m}, \in, A)\models\varphi\implies (V_{\lambda+m}, \in, A\cap V_\lambda)\models\varphi.$$ For example, the $\Pi^1_1$-indescribable cardinals are exactly the weakly compact cardinals. (Note that we can go beyond $\Pi^m_n$-describability - quite a ways, even - but let's ignore that for now.)
I'm interested in a version of indescribability where we pay attention to what can happen in generic extensions of $V$ (I don't think this really qualifies as a "generic" version indescribability, but there's a vague connection). Specifically, for an ordinal $\alpha$ and a set $C\subseteq V_\alpha$, write "$(V_\alpha, \in, C)\models_f\varphi$" if $((V_\alpha)^{V[G]}, \in, C)\models\varphi$ for every $G$ which is set-generic over $V$ (note that the forcing for which $G$ is generic need not be an element of $V_\alpha$). Now say that a cardinal $\kappa$ is $\Pi^m_n$-forcing indescribable if for every $\Pi_n$-sentence $\varphi$ in the language of set theory with a unary predicate and every $A\subseteq V_\kappa$, there is some $\lambda<\kappa$ such that $$(V_{\kappa+m}, \in, A)\models_f\varphi\implies (V_{\lambda+m}, \in, A\cap V_\lambda)\models_f\varphi.$$ Note that $A$ is a fixed set in the ground model.
My question is, roughly how "big" are the forcing-indescribable cardinals? For example, it is not clear to me what the relationship is between $\Pi^1_1$-indescribable (= weakly compact) cardinals and $\Pi^1_1$-forcing indescribable cardinals. I believe that in $L$, $\Gamma$-forcing indescribability implies $\Gamma$-indescribability (since we can replace $\varphi$ with "$\varphi$ holds in $L$"), but I don't see the converse, or how this holds in general $V$.
 A: Let me start things off by providing an upper bound. The bound is
very large, however, and I expect that it can be improved, perhaps
dramatically. But at least it shows the consistency of your large
cardinal relative to some other well-studied large cardinals.
Theorem. If $\kappa$ is $1$-$C^{(2)}$-extendible, then it is
forcing $\Pi^m_n$-indescribable for every $m,n$.
Definition. A cardinal $\kappa$ is
$1$-$C^{(2)}$-extendible,
if there is an elementary embedding $j:V_{\kappa+1}\to
V_{\theta+1}$, with critical point $\kappa$, such that the target
$j(\kappa)=\theta$ is $\Sigma_2$-correct in $V$, meaning
$V_\theta\prec_{\Sigma_2} V$.
This is a fairly strong large cardinal notion, far stronger than
the totally indescribable cardinals you mention in your question.
For example, every 1-extendible cardinal is superstrong and much
more. But these cardinals are weaker than Vopěnka's principle.
Proof. Assume that $\kappa$ is $1$-$C^{(2)}$-extendible. So
there is an elementary embedding $j:V_{\kappa+1}\to V_{\theta+1}$
with critical point $\kappa$ and $j(\kappa)=\theta$ is
$\Sigma_2$-correct in $V$.
Suppose now that $A\subset V_\kappa$ and $\langle
V_{\kappa+m},\in,A\rangle\models_f\varphi$, which means that
$\langle V[G]_{\kappa+m},\in,A\rangle\models\varphi$ for every
forcing extension $V[G]$. This is a $\Sigma_2$ property about $A$
and $\kappa$, since any violation of it would be revealed inside
some large enough $V_\eta$, using forcing inside that $V_\eta$.
Thus, by $\Sigma_2$-correctness, we see that $V_\theta$ agrees that
$\langle V_{\kappa+m},\in,A\rangle\models_f\varphi$. Since
$A=j(A)\cap\kappa$, we may pull this back by elementarity to
conclude that there is some $\lambda<\kappa$ with $\langle
V_{\lambda+m},\in,A\cap V_{\lambda+m}\rangle\models_f\varphi$
inside $V_\kappa$. But $\kappa$ itself must also be
$\Sigma_2$-correct, and so actually $\langle
V_{\lambda+m},\in,A\cap V_{\lambda+m}\rangle\models_f\varphi$ in
$V$, as desired. QED
I'll think some more about lower bounds and about pulling down the
strength of the hypothesis.
Update. I've realized that we can improve the upper bound as
follows. We don't really need the "$+1$", since that actually
provided a uniform version of the phenomenon, with the same
embedding working for every $A$.
Thomas Johnstone and I defined that a cardinal $\kappa$ is
uplifting, if it is inaccessible and $V_\kappa\prec V_\theta$ for
cofinally many inaccessible cardinals $\theta$. (J. D. Hamkins, T.
Johnstone, Resurrection axioms and uplifting
cardinals)
A boldface version is that $\kappa$ is strongly uplifting, if for
every $A\subset V_\kappa$ there are cofinally many inaccessible
cardinals $\theta$ for which $\langle
V_\kappa,\in,A\rangle\prec\langle V_\theta,\in,A^*\rangle$ for some
$A^*\subset V_\theta$. (J. D. Hamkins, T. Johnstone, Strongly
uplifting cardinals and boldface
resurrection)
These have diverse equivalent formulations, as I mention on the
linked blog post, connected with strengthenings of the strongly
unfoldable cardinals.
Let me now strengthen this a little more, for the present
application, with the following new large cardinal concept.
Definition. A cardinal $\kappa$ is strongly
$C^{(n)}$-uplifting, if for every $A\subset V_\kappa$ there is a
$\Sigma_n$-correct cardinal $\theta$ and $A^*\subset V_\theta$ with
$\langle V_\kappa,\in,A\rangle\prec\langle
V_\theta,\in,A^*\rangle$.
This is what we really needed in the theorem above.
Theorem. If $\kappa$ is strongly $C^{(2)}$-uplifting, then it
is forcing $\Pi^m_n$-indescribable for every $m$ and $n$.
Proof. Argue as in the first theorem above, but now we have
only $\langle V_\kappa,\in,A\rangle\prec\langle
V_\theta,\in,A^*\rangle$, instead of $j$. If $\langle
V_\kappa,\in,A\rangle+m\models_f\varphi$, then this will be true
inside $V_\theta$ since it is $\Sigma_2$-correct, and so $V_\theta$
thinks that this holds on an initial segment of $A^*$, and so we
get $\lambda<\kappa$ with $\langle V_\lambda,\in,A\cap
V_\lambda\rangle\models_f\varphi$ inside $V_\kappa$, which is right
about this since $\kappa$ is itself $\Sigma_2$-correct. QED
I think the strongly $C^{(2)}$-uplifting cardinals are
comparatively weak, and absolute to $L$, but I'll think more about
it.
