Rational-relative-to-real groups are frequently called inverse semi-rational groups (or cut groups). This term was Inverse semi-rational groups are a subclass of semi-rational groups. A finite group $G$ is semi-rational if for every $g \in G$ there exists a positive integer $m_0$ such that every primitive power $g^a$ ($a \in (\mathbb{Z}/\operatorname{order}(g))^\times$) is conjugate to $g$ or $g^{m_0}$. A semi-rational group is inverse semi-rational if we can take $m_0 = -1$ for every $g \in G$ in the above definition. The terms semi-rational groups and inverse semi-rational groups were 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 trialtytriality groups are discarded in the proof, so should actually be removed from the list in Theorem 1.1.
The simple inverse semi-rational groups were derived in Theorem 5.1 of [Bächle-Caicedo-Jespers-Maheshwary, Global and local properties of finite groups with only finitely many central units in their integral group ring, Journal of Group Theory 24, 2021, 1163-1188].
