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We report on the detailed composition of ternary GaAsP nanowires (NWs) grown using self-catalyzed vapor-liquid-solid (VLS) growth by molecular beam epitaxy. We evidence the formation of an unintentional shell, which enlarges by vapor-solid growth concurrently to the main VLS-grown core. The NW core and unintentional shell have typically different chemical compositions if no effort is made to adjust the growth conditions. The compositions can be made equal by changing the substrate temperature and the P/As flux ratio in the vapor phase. In all cases, we still observe the existence of a P-rich interface between the GaAsP NW core and the unintentional shell, even if favorable growth conditions are used.