Rational-relative-to-real groups are frequently called semi-rational groups.  This term was introduced in the article [Chillag-Dolfi, Semi-rational solvable groups, *Journal of Group Theory* **13**(4), 2010, 535-548].  In [Alavi-Daneshkhah, On semi-rational finite simple groups, *Monatshefte für Mathematik* **184**(2), 2017, 175-184] the authors determined the finite simple semi-rational groups.

Note that in the statement of Theorem 1.1 the group $G_2(4)$ is missing.  The Tits group, the Suzuki groups, the Ree groups and the Steinberg trialty groups are discarded in the proof, so should actually be removed from the list in Theorem 1.1.