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 of Alavi–Daneshkhah, 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.
