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 $qoverline q$ 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 $gamma prightarrow X^+n$ 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.
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