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.

- $\mathbb{P}$ preserves stationary subsets of $\omega_1$
- $\mathbb{P}$
*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. - 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).