I'm working with the following variant of Baumgartner's forcing to add a club subset of $\omega_1$, talked about by Mitchell on page 3 here: https://arxiv.org/pdf/math/0407225.pdf
Let $S \subseteq \omega_1$ be stationary. The poset $\mathbb{P}_S$ is defined by:
Conditions are ordered pairs $p = (I_p, O_p)$ where $I_p \subseteq S$ is finite and $O_p$ consists of finitely many intervals of the form $(\alpha, \beta]$ where $\alpha < \beta < \omega_1$ such that $I_p \cap (\alpha, \beta] = \emptyset$ whenever $(\alpha, \beta] \in O_p$.
Ordering is defined as follows: $p \leq q$ iff $I_p \supseteq I_q$ and $O_p \supseteq O_q$.
$\mathbb{P}_S$ adds a subset of $\omega$, but I do not see how to generically code such a real. Can anyone help shed some light on this?