  In this Letter, we report compressible mode effects on relativistic
magnetohydrodynamic (RMHD) turbulence in Poynting-dominated plasmas using
3-dimensional numerical simulations. We decomposed fluctuations in the
turbulence into 3 MHD modes (fast, slow, and Alfv\'en) following the procedure
mode decomposition in (Cho & Lazarian 2002), and analyzed their energy spectra
and structure functions separately. We also analyzed the ratio of compressible
mode to Alfv\'en mode energy with respect to its Mach number. We found the
ratio of compressible mode increases not only with the Alfv\'en Mach number but
with the background magnetization, which indicates a strong coupling between
the fast and Alfv\'en modes and appearance of a new regime of RMHD turbulence
in Poynting-dominated plasmas where the fast and Alfv\'en modes strongly
couples and cannot be separated, different from the non-relativistic MHD case.
This finding will affect particle acceleration efficiency obtained by assuming
Alfv\'enic critical balance turbulence, and will change the resulting photon
spectra by non-thermal electrons.
