  We study the decays of the tau-sleptons (stau_{1,2}) and tau-sneutrino
(snu_tau) in the Minimal Supersymmetric Standard Model (MSSM) with complex
parameters A_tau, mu and M_1 (U(1) gaugino mass). We show that the effect of
the CP phases of these parameters on the branching ratios of stau_{1,2} and
snu_tau decays can be quite strong in a large region of the MSSM parameter
space. This could have an important impact on the search for stau_{1,2} and
snu_tau and the determination of the MSSM parameters at future colliders.
