Polyelectrolyte complex coacervation plays an essential role in compartmentalization in living cells and is important in multiple scenarios of prebiotic evolution. This has fueled considerable interest in associative phase separation, which occurs in solutions of oppositely charged biological polyelectrolytes, including those containing double-stranded DNA. The high bending rigidity of DNA duplexes has an effect on the properties of the polymer-rich phase. In this paper, we combine scaling approaches and the random phase approximation (RPA) to develop a theory of coacervates formed from semiflexible (rigid) polyanions and flexible polycations. At low stiffness of the polyanion, coacervates are isotropic liquids with two different correlation lengths, equal to the mesh sizes of the polyanion and polycation interpenetrating semidilute solutions. When the polyanion stiffness exceeds a threshold, the coacervate undergoes liquid crystalline ordering (LCO). The formation of a nematic phase is induced by anisotropic excluded volume and anisotropic Coulomb interactions between semiflexible polyanions. Our theoretical predictions of LCO within the coacervate formed from flexible and semiflexible polyelectrolytes are consistent with experimental studies.