Fix a countable transitive model $M$ and consider the collection $\mathbb{P}$ of all forcing extensions of $M$ (i.e., the generic multiverse of $M$), ordered by reverse containment. What happens if we force with the partial order $\mathbb{P}$? Does it collapse $\omega_1$ or $\mathfrak{c}$? Is it equivalent to $\mathrm{Col}(\omega,\omega_1)$ or some other well-known poset? What kind of set theory does the union of all models in the generic filter satisfy? Here it is mentioned that $\mathbb{P}$ is almost (but not exactly) countably closed.
I remember seeing a similar question about forcing with the poset of all posets (up to size $\kappa$?) under complete embedding, but can't find it now