If we suppose [SU(5) × U(1)A]/Z5 is a subgroup of SO(10)F, then we have the intermediate scale symmetry breaking [SO(10)F × U(1)B]/Z4 → [SU(5) × U(1)χ]/Z5 where In that case, note that the Standard Model fermion fields (including the right handed neutrinos) come from all three [SO(10)F × U(1)B]/Z4 representations.
This is a standard practice in the GUT model building literature anyway).
Higgs field, we can have VEVs which breaks the GUT group down to [SU(5) × U(1)χ]/Z5.
And we can always introduce a sterile neutrino φ which is invariant under [SO(10) × U(1)B]/Z4 and add the Yukawa coupling OR we can add the nonrenormalizable term Either way, the 10 component of the fermion 161 gets taken care of so that it is no longer chiral.
One reason for studying flipped SO(10) is because it can be derived from an E6 GUT model.