By an Aronszajn tree, I mean a tree of height $\omega_1$ with countable levels and no branch. Such a tree is Souslin if it has no uncountable antichains and special if it can be written as the countable union of antichains. It is well known that there is both a ccc forcing and a $\sigma$-closed forcing for adding Souslin trees and, given any Aronszajn tree $T$ (Souslin or otherwise) there is a ccc forcing for specializing $T$. Therefore we can add Souslin trees, and then specialize them, giving us new special Aronszajn trees. In this post I have a variety of questions about forcings which add Aronszajn trees that are neither special nor Souslin. The most basic of these is as follows.
Question 1: What are some of the most common ways of forcing to add an Aronszajn tree which is neither Souslin nor special? In this post Mohammad Golshani cites a theorem that immediately implies that such trees exist under certain $\diamondsuit$ hypotheses. Therefore forcings to add diamonds suffice but I'm curious whether there is a more explicit forcing where the generic is actually the tree.
My next question is about killing Souslin-ness (or whatever it's called) without specializing.
Question 2: Suppose $T$ is Souslin. Does there exist a forcing notion $\mathbb P$ such that in $V^\mathbb P$, $T$ is still Aronszajn but no longer Souslin and also not special?
I'm also curious about these questions in the case of fat Aronszajn trees: Aronszajn trees with the requirement of countable levels removed. The same ccc specializing forcing works to make such trees special so under MA all such trees are special. I'm curious whether non-special fat Aronszajn trees can be built by hand using forcing.
Question 3: What forcings will add a fat, non-special Aronszajn tree?