  The occurrence of non-Abelian symmetry-locked states in ultracold fermionic
mixtures with four components is investigated. 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 is 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 (TSFL) phase. This phase is present
also for not too large repulsive intra-pair 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
$^{171}\mathrm{Yb}$-$^{173}\mathrm{Yb}$, indicating that the TFSL phase can be
achieved also without tuning the interactions between $\mathrm{Yb}$ atoms.
