  The dominant contributions to W-+ H+- production at the LHC are the
tree-level b anti-b annihilation and the gg fusion. We perform for the case of
the complex MSSM a complete calculation of the NLO EW corrections to the b
anti-b annihilation channel and a consistent combination with other
contributions including the standard and SUSY QCD corrections and the gg
fusion, with resummation of the leading radiative corrections to the
bottom-Higgs couplings and the neutral Higgs-boson propagators. We observe a
large CP-violating asymmetry, arising mainly from the gg channel.
