  Ultra-peripheral collisions (UPCs) of relativistic ions are an important tool
for studying photoproduction at high energies. Vector meson photoproduction is
an important tool for nuclear structure measurements and other applications. A
future electron-ion collider (EIC) will allow additional studies, using virtual
photons with a wide range of $Q^2$. We propose a significant expansion of the
UPC and EIC photoproduction physics programs to include charged final states
which may be produced via Reggeon exchange. We consider two examples:
$a_2^+(1320)$, which is a conventional $q\overline q$ meson, and the exotic
$Z_c^+(4430)$ state (modeled here as a tetraquark). The $Z_c^+(4430)$
cross-section depends on its internal structure, so photoproduction can test
whether the $Z_c^+(4430)$ is a tetraquark or other exotic object. We calculate
the rates and kinematic distributions for $\gamma p\rightarrow X^+n$ in $pA$
UPCs and $ep$ collisions at an EIC and in UPCs. The rates are large enough for
detailed studies of these final states. Because the cross-section for Reggeon
exchange is largest near threshold, the final state rapidity distribution
depends on the beam energies. At high-energy colliders like the proposed LHeC
or $pA$ collisions at the LHC, the final states are produced at far forward
rapidities. For lower energy colliders, the systems are produced closer to
mid-rapidity, within reach of central detectors.
