We show that large-amplitude, non-planar, Alfvén-wave (AW) packets are exact nonlinear solutions of the relativistic magnetohydrodynamic 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 Alfvén 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.
