Suppose $\mathbb{P}$ is a forcing with the following properties: Let $G \subseteq \mathbb{P}$ be filter generic over $V$, then there exists $A \in V[G]$ such that $V[G]$ thinks $A$ is countable and $A \subseteq {}^\omega 2 \cap V$, but $A$ is not covered by any ground model countable set. That is, in the generic extension there is a countable set of ground model reals that can not be covered by a countable set from the ground model.

Certainly $\mathbb{P}$ is not a proper forcing.

Is it possible that $\mathbb{P}$ can preserve $\aleph_1$? As $A$ need not be a set in $V$, it seems at least theoretically possible that $\mathbb{P}$ does not need to collapse any cardinals.

Now I would like to iterate this forcing by itself. However, since it is not proper, I do not know what properties can be expected to be preserved.

Although this may be a bit too open ended, my next question is : What classes of forcing (by which I means things like c.c.c., proper, semi-proper, etc) could $\mathbb{P}$ potentially belong to given that it adds a countable set of ground model real that can not be covered by a countable ground model set. (Certainly $\mathbb{P}$ can not be c.c.c. or proper.)

Do any of these classes have preservation theorems for iterations (countable support or perhaps some other type of iterations)? I am most interested in perserving $\aleph_1$, preserving ${}^\omega\omega$-bounding, or $\aleph_2$-chain conditions. If it is applicable, one may assume $\mathbb{P}$ has size $\aleph_1$ if $\mathsf{CH}$ holds.

I am looking for some class of forcing that can help handle iterations of non-proper forcings like $\mathbb{P}$. Thanks for any information that can be provided.