  We report a measurement of the $\pi^-$ photoproduction beam asymmetry for the
reaction $\vec{\gamma} p \rightarrow \pi^- \Delta^{++}$ using data from the
GlueX experiment in the photon beam energy range 8.2--8.8 GeV. The asymmetry
$\Sigma$ is measured as a function of four-momentum transfer $t$ to the
$\Delta^{++}$ and compared to phenomenological models. We find that $\Sigma$
varies as a function of $t$: negative at smaller values and positive at higher
values of $|t|$. The reaction can be described theoretically by $t$-channel
particle exchange requiring pseudoscalar, vector, and tensor intermediaries. In
particular, this reaction requires charge exchange, allowing us to probe pion
exchange and the significance of higher-order corrections to one-pion exchange
at low momentum transfer. Constraining production mechanisms of conventional
mesons may aid in the search for and study of unconventional mesons. This is
the first measurement of the process at this energy.
