  The observed galaxy distribution via galaxy redshift surveys appears
distorted due to redshift-space distortions (RSD). While one dominant
contribution to RSD comes from the Doppler effect induced by the peculiar
velocity of galaxies, the relativistic effects, including the gravitational
redshift effect, are recently recognized to give small but important
contributions. Such contributions lead to an asymmetric galaxy clustering along
the line of sight, and produce non-vanishing odd multipoles when
cross-correlating between different biased objects. However, non-zero odd
multipoles are also generated by the Doppler effect beyond the distant-observer
approximation, known as the wide-angle effect, and at quasi-linear scales, the
interplay between wide-angle and relativistic effects becomes significant. In
this paper, based on the formalism developed by Taruya et al., we present a
quasi-linear model of the cross-correlation function taking a proper account of
both the wide-angle and gravitational redshift effects, as one of the major
relativistic effects. Our quasi-linear predictions of the dipole agree well
with simulations even at the scales below $20\,h^{-1}\,$Mpc, where
non-perturbative contributions from the halo potential play an important role,
flipping the sign of the dipole amplitude. When increasing the bias difference
and redshift, the scale where the sign flip happens is shifted to a larger
scale. We derive a simple approximate formula to quantitatively account for the
behaviors of the sign flip.
