  In this paper, we report our detailed analysis of the new strong-coupling
regime between Alfv\'en and fast modes in Poynting-dominated plasma turbulence,
reported in our previous work Takamoto & Lazarian (2016), which is an important
effect for many relativistic plasma phenomena, and calls for new theories of
Poynting-dominated MHD turbulence. We performed numerical simulations of
relativistic MHD turbulence in isothermal plasmas, and analyzed the ratio of
fast to Alfv\'en mode energy. We found that the increase of the fast mode with
the background $\sigma$-parameter can be observed even in isothermal plasma,
showing that such a phenomena is universal in trans-Alfv\'enic turbulence in
Poynting-dominated plasmas. To study the detailed energy conversion process, we
also performed a series of simulations of decaying turbulence injecting pure
Alfv\'en, fast, and slow modes, respectively, and investigated the development
of the mode conversion from the each mode. We also found that the mode
conversion between Alfv\'en and fast modes is nearly insensitive to the
background temperature. Finally, we report a result of a simulation with
initially fast mode dominated turbulence. It developed into a temporally
strong-coupling regime, which is a strong evidence for the existence of our
suggesting strong-coupling regime of fast and Alfv\'en modes. Our result
suggests that the strong turbulence in Poynting-dominated plasma is very
different from that in the non-relativistic plasma. It will also give an
important guidance to studies of particle acceleration and non-thermal photon
emission from Poynting-dominated plasma.
