Are the $\mathsf{HOD}$s preserved by weakly homogeneous forcings? We saw the following theorem in class: > Let $M$ be a transitive model of $\mathsf{ZFC}$, let $\Bbb P\in M$ be a weakly homogeneous partially ordered set, let $G$ be $\Bbb P$-generic over $M$ and let $A$ be a class in $M[G]$ with $A\subseteq M$. Then $\mathsf{HOD}_A^{M[G]}\subseteq M$. By the question linked in the comments we have $\mathsf{HOD}_A^{M[G]}\subseteq\mathsf{HOD}_{A\cup\\{\Bbb P\\}}^M$, is there a simple example that shows that this inclusion can be strict? are there natural assumptions on $\Bbb P$ that guarantee an equality instead of an inclusion?

Here's an example with $A=\varnothing$.

Start with $L$. Then there is a Suslin tree which is rigid there. Force with it to generate $V=L[A]$. By rigidity, the branch added is a unique branch in the tree, so it is ordinal definable (since the tree was in $L$). In particular, $V=\mathrm{HOD}$.

Now force with $\Bbb P=\operatorname{Col}(\omega,\omega_1)$. Then $V[G]=L[G]$, and so $\mathrm{HOD}^{V[G]}=L$, since $\Bbb P$ is in fact in $L$ and homogeneous there. Therefore we have that:

$$\mathrm{HOD}^{V[G]}=L\subsetneq V=\mathrm{HOD}^V=\mathrm{HOD}^V_{\\{\Bbb P\\}}.$$