  Exclusive photoproduction of a $\gamma\,\pi^\pm$ pair in the kinematics where
the pair has a large invariant mass and the final nucleon has a small
transverse momentum is described in the collinear factorization framework. The
scattering amplitude is calculated at leading order in $\alpha_s$ and the
differential cross sections for the process are estimated in the kinematics of
the JLab~12-GeV experiments. The order of magnitude of the predicted
cross-sections seems sufficient for a dedicated experiment to be performed. The
process turns out to be very sensitive to the axial generalized parton
distribution combination $\tilde{H}_u - \tilde{H}_d$.
