In a comment to Miha's question in Forcing PFA with ccc forcing, I suggested that if such situation is even possible, it might be achieved by screwing with PFA by some ccc forcing (e.g. adding a Cohen real), and then "fixing all the problems" via a ccc forcing.

And that raises the question:

Apart of adding a Cohen real, what sort of ccc forcing adds a Suslin tree on $\omega_1$?

Of course, take any ccc forcing $\Bbb P$, and consider $\Bbb P\times\operatorname{Add}(\omega,1)$ is an answer. So to avoid this triviality, we only consider forcings which do not add a Cohen real themselves. So we can rephrase the question:

Suppose that $\Bbb P$ is a ccc forcing which adds a Suslin tree on $\omega_1$. Does $\Bbb P$ add a Cohen real?

If the answer is negative (namely it is possible to add a new Suslin without adding a Cohen), does the answer change if we assume the ground model has no Suslin trees?