What are some examples of posets $\mathbb{P}$ which have the following properties? It's OK if the definition uses large cardinals or some other hypothesis.


1. $\mathbb{P}$ preserves stationary subsets of $\omega\_1$
2. $\mathbb{P}$ *fails* to have the $\omega$-covering property. So in particular, $\mathbb{P}$ cannot be $\sigma$-distributive, or proper on any stationary set of countable models.
3. Whenever $\kappa$ has uncountable cofinality, then this remains true in $V^{\mathbb{P}}$.


Item (3) rules out Namba forcing,Prikry forcing, and the standard stationary tower forcings with critical point at least $\omega\_2 $ (the latter preserve stationary subsets of $\omega\_1 $ if the height of the tower is a Woodin cardinal, but change lots of regular cardinals to cof $\omega $).


Item (2) rules out several forms of antichain sealing forcings (all variants of sealing forcings I know of are of the form "proper followed by shooting a club", which has the countable covering property). 



Posets preserving stationary subsets of $\omega_1$ and no new $\text{cof}(\omega)$ ordinals, but without countable covering property
Let $(\kappa\_n: n<\omega)$ be an increasing sequence of measurable cardinals with limit $\kappa,$ and for each $n$ let $U\_n$ be a normal measure on $\kappa\_n.$ Let $\mathbb{P}$ be the Prikry type forcing notion which adds one element Prikry to each $\kappa\_n:$


$p\in \mathbb{P}$ iff $p=(s, A),$ where 


1)$s$ is a finite sequence,


2) $\forall i< |s|, s(i) < \kappa\_i,$


3) $A=(A\_n: |s| \leq n < \omega),$ and $A\_n \in U\_n$,


$p=(s,A)\leq q=(t, B)$ iff:


1) $s$ end extends $t$,


2) $\forall |t| \leq i < |s|, s(i)\in B\_i,$


3) $\forall |s| \leq n < \omega, A\_n \subseteq B\_n.$


The forcing adds an $\omega$-sequence to $\kappa,$ and can not be covered by any ground model set of size $\omega$ (in fact of size $< \kappa$), it adds no bounded subsets to $\kappa,$ and does not change cofinalities. 


So the forcing is as requested.


