  We report on a dynamical mean-field theoretical analysis of emerging
low-temperature phases in multicomponent gases of fermionic
alkaline-earth(-like) atoms in state-dependent optical lattices. Using the
example of $^{173}$Yb atoms, we show that a two-orbital mixture with two
nuclear spin components is a promising candidate for studies of not only
magnetic but also staggered orbital ordering peculiar to certain solid-state
materials. We calculate and study the phase diagram of the full Hamiltonian
with parameters similar to existing experiments and reveal an antiferroorbital
phase. This long-range-ordered phase is inherently stable, and we analyze the
change of local and global observables across the corresponding transition
lines, paving the way for experimental observations. Furthermore, we suggest a
realistic extension of the system to include and probe a Jahn-Teller source
field playing one of the key roles in real crystals.
