Preservation of stationary sets by Mitchell forcing quotients It is well-known that Mitchell's forcing $\mathbb M$ for the tree property at $\omega_2$, which turns a weakly compact $\kappa$ into $\omega_2$ while adding many reals, is a projection of adding $\kappa$-many Cohen reals times a countably closed forcing.  Call this $\mathbb C \times \mathbb T$.
Suppose $G \subseteq \mathbb M$ is generic and $X \subseteq \omega_1$ is stationary in $V[G]$.  Is the stationarity of $X$ preserved by $(\mathbb C \times \mathbb T)/G$?
 A: In this paper, Gilton and Krueger show that the failure of approachability at $\omega_2$ is implied by the existence of a disjoint stationary sequence, which is a disjoint sequence of sets $\langle S_\alpha : \alpha \in T \rangle$, where $T \subseteq \omega_2 \cap \mathrm{cof}(\omega_1)$ is stationary and each $S_\alpha$ is a stationary subset of $[\alpha]^\omega$.  It follows from the arguments in their paper that Mitchell's forcing adds a disjoint stationary sequence.  They key is that the quotient forcings over initial segments are proper (projections of ccc x countably closed, just like the whole forcing).
Now in the extension by $\mathbb C \times \mathbb T$ like in the question, the approachability property holds because CH holds in an inner model with the same cardinals (the extension by $\mathbb T$).  Thus the passage from $V^{\mathbb M}$ to $V^{\mathbb C \times \mathbb T}$ must kill the disjoint stationary sequence $\langle S_\alpha : \alpha \in T \rangle$.  Since it is $\kappa = \omega_2$-c.c., it cannot kill the stationarity of $T$.  Thus it must kill almost all of the $S_\alpha$.  Since they are stationary subsets of countable subsets of ordinals of size $\omega_1$, they correspond to stationary subsets of $\omega_1$, so the quotient kills many stationary subsets of $\omega_1$.
EDIT: A more direct argument, without using all of the Gilton-Krueger work.  The stationary subset of $[\alpha]^\omega$ referred to above is just
$$([\alpha]^\omega)^{V[G \restriction \alpha][c_\alpha]} \setminus  ([\alpha]^\omega)^{V[G \restriction \alpha]},$$
where $c_\alpha$ is the Cohen real added at stage $\alpha$.  By a theorem of Gitik, this is stationary, and its stationarity is preserved by the tail forcing.  But in the model $V^{\mathbb T}$, there is a club in $[\alpha]^\omega$ consisting of countable sets from $V$.
