We report a novel phase separation phenomenon observed in the growth of ternary In(x)Ga(1-x)As nanowires by metalorganic chemical vapor deposition. A spontaneous formation of core-shell nanowires is investigated by cross-sectional transmission electron microscopy, revealing the compositional complexity within the ternary nanowires. It has been found that for In(x)Ga(1-x)As nanowires high precursor flow rates generate ternary In(x)Ga(1-x)As cores with In-rich shells, while low precursor flow rates produce binary GaAs cores with ternary In(x)Ga(1-x)As shells. First-principle calculations combined with thermodynamic considerations suggest that this phenomenon is due to competitive alloying of different group-III elements with Au catalysts, and variations in elemental concentrations of group-III materials in the catalyst under different precursor flow rates. This study shows that precursor flow rates are critical factors for manipulating Au catalysts to produce nanowires of desired composition.