TITLE: Forcing notions adding minimal reals
QUESTION [12 upvotes]: I am looking for a comprehensive list of known forcing notions which add a minimal real into the ground model. I know some of them like the Sacks forcing, or the Judah-Shelah's example of a c.c.c. forcing which adds a minimal real (and a few more).

Question. What are other known (or less known) examples of forcing notions which add a minimal generic real?

REPLY [11 votes]: Splitting forcing is a little-known forcing that adds a splitting real and creates a minimal extension. It consists of splitting trees ordered by inclusion. A splitting tree is a perfect tree $T \subseteq 2^{<\omega}$ where for every $s \in T$ there is $m \in \omega$ so that for all $n \geq m$, $i \in 2$, there is $t \in T$, $s \subseteq t$ with $t(n)= i$.
In my recent preprint (arxiv link), I show that splitting forcing adds a minimal real (Corollary 4.20) and provide part of an argument for the minimality of the forcing extension (Corrolary 3.21).
Other than that, splitting forcing is $\omega^\omega$-bounding but as far as I know it is unknown whether it has the Sacks-property (see also Question 6.4 in On splitting trees by Laguzzi, Mildenberger and Stuber-Rousselle).