  We show that large-amplitude, non-planar, Alfv\'en wave (AW) packets are
exact nonlinear solutions of the relativistic MHD equations when the total
magnetic-field strength in the local fluid rest frame ($b$) is a constant. We
derive analytic expressions relating the components of the fluctuating velocity
and magnetic field. We also show that these constant-$b$ AWs propagate without
distortion at the relativistic Alfv\'en velocity and never steepen into shocks.
These findings and the observed abundance of large-amplitude, constant-$b$ AWs
in the solar wind suggest that such waves may be present in relativistic
outflows around compact astrophysical objects.
