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. 



Preservation Results for Iterations of Non-Proper Forcing
The existence of a forcing notion $\mathbb{P}$ like that is equivalent to $\neg\text{CH}$. 


On the one hand, Noah's comment shows that if CH holds, then one cannot add a countable set of ground-model reals that is not covered by any countable ground model set, while preserving $\omega\_1$.


Conversely, suppose that CH fails in $V$, and consider Namba forcing. This adds a new cofinal $\omega$-sequence in $\omega\_2^V$, and hence because $\neg\text{CH}$ it adds a countable set of ground-model reals, which is not covered by any ground-model countable set, or indeed by any ground model set of size $\omega\_1$ in the ground model, but it preserves $\omega\_1$. QED


It seems to be consistent relative to large cardinals that [Namba forcing is semi-proper](https://mathoverflow.net/q/165105/1946), in which case one might hope to iterate it.


Here is another kind of example. Start in $V$ with a measurable cardinal $\kappa$, and let $V[G]$ be the extension obtained by adding $\kappa$ many Cohen reals, so that $2^\omega=\kappa$ in $V[G]$. Now, in $V[G]$, let $\mathbb{P}$ be Prikry forcing for a normal measure $\mu$ in $V$, as defined in $V$. If $H\subset\mathbb{P}$ is $V[G]$-generic, then we can view $V[G][H]$ as $V[H][G]$, that is, as first doing Prikry forcing, and then adding reals. So all cardinals are preserved. But the Priky sequence will add a new $\omega$-sequence cofinal in $\kappa$, and hence a new countable set of reals from $V[G]$, which is not covered by any set in $V[G]$ of size less than $\kappa$ in $V[G]$. Thus, forcing over $V[G]$ with the Prikry forcing of $V$ has your desired properties. 


And there are various complicated ways to iterate Priky forcing, depending on what you want to accomplish, and what kind of large cardinals you have available. 


