Based on the linear approximation of liquid constitutive equation, this paper investigated the instability and breakup of an annular power-law liquid jet into bounded and compressible gas medium, with both para-sinuous mode and para-various mode disturbances. The influences of gas-liquid velocity difference, Mach number, outer gas confinement, and gas-liquid density ratio were studied in detail. Numerical results show that with the increase of gas-liquid velocity difference, the jet instability is gradually dominated by the Kelvin-Helmholtz wave. The annular jet of power-law fluids in the bounded gas stream is more unstable than that in the open gas medium. As the gas velocity increases, the gas confinement has less impact on the jet instability. The gas compressibility can effectively destabilize the annular jet of power-law fluids and decrease the breakup scale. In low-speed gas stream, the jet breakup is mainly dominated by the para-sinuous mode. With the increase of the gas density and velocity as well as theMach number, the para-varicose mode also exerts a significant effect on the breakup of power-law liquid jets. Especially in the cases of largeMach number and high gas velocity, either disturbance mode has the potential to dominate the breakup of power-law liquid jets.