The realization of experiments in ultracold multicomponent mixtures, also involving more atomic species, opened the way to the study of exotic quantum phases and unconventional superfluidity, as, for instance non-Abelian superfluid phases. In this paper we study the occurrence of non-Abelian symmetry-locked superfluid states in ultracold fermionic mixtures with four components, showing that such states can be studied in current day experiments with Yb171-Yb173 isotopes. We study the phase diagram in the presence of an attractive interaction between the species of two pairs of the mixture, and general (also repulsive) interactions between the species of each pair. This system can be physically realized, e.g., in mixtures of two different earth-alkaline species, both of them with two hyperfine levels selectively populated. We find an extended region of the diagram exhibiting a two-flavors superfluid symmetry-locking (TFSL) phase. The locking corresponds to the presence of a order parameter involving - in all the possible and distinct permitted ways - two fermions, one of them belonging to the first pair and the second to the other one. This TSFL phase is present also for not too large repulsive intrapair interactions and it is characterized by a global non-Abelian symmetry group obtained by locking together two independent invariance groups of the corresponding normal state. Explicit estimates are reported for the mixture of the fermionic isotopes Yb171-Yb173, indicating that the TFSL phase can be achieved also without tuning the interactions between Yb atoms.
