  The interference between Compton scattering and nonlinear Compton scattering
from a two-color field in the X-ray regime is theoretically examined for bound
electrons. The underlying phase shifts are analysed using a perturbative
approach in the incoming classical field. The perturbative approach is bench
marked with a non-perturbative approach in the classical field. The
interference for different combinations of linear polarization of the two
fields is examined when the Compton and the nonlinear Compton scattered waves
have the same wave vector and polarization. Only two cases exhibit
interference. When there is interference, the calculations reveal an intrinsic
phase difference between the Compton scattered wave function and the nonlinear
Compton scattered wave function of either 0 or $\pi$ depending on the
scattering angle.
