We consider the Euler equations for the incompressible flow of an ideal fluid
with an additional rough-in-time, divergence-free, Lie-advecting vector field.
In recent work, we have demonstrated that this system arises from Clebsch and
Hamilton-Pontryagin variational principles with a perturbative geometric rough
path Lie-advection constraint. In this paper, we prove local well-posedness of
the system in $L^2$-Sobolev spaces $H^m$ with integer regularity $m\ge \lfloor
d/2\rfloor+2$ and establish a Beale-Kato-Majda (BKM) blow-up criterion in terms
of the $L^1_tL^\infty_x$-norm of the vorticity. In dimension two, we show that
the $L^p$-norms of the vorticity are conserved, which yields global
well-posedness and a Wong-Zakai approximation theorem for the stochastic
version of the equation.