  The emergent phases of strongly correlated spin-1/2 Fermi gases of Rydberg
dressed atoms in a one dimensional optical lattice are theoretically
investigated. At weak coupling a bosonization description is used to
demonstrate the ability to drive alternating quantum phase transitions between
distinct Luttinger liquids. At strong coupling the ground state develops
non-trivial phase separation exhibiting Luttinger liquid ''puddles'' separated
by magnetic domain walls due to the interplay of the incommensurate filling and
the Rydberg core length scale. These phases can be detected in ultracold gases
of Rydberg atoms made from $^6$Li.
