Geometric phases appear ubiquitously in many and diverse areas of the physical sciences, ranging from classical and molecular dynamics to quantum mechanics and solid-state physics. In the realm of optics, similar phenomena are known to emerge in the form of a Pancharatnam-Berry phase whenever the polarization state traces a closed contour on the Poincaré sphere. While this class of geometric phases has been extensively investigated in both free-space and guided wave systems, the observation of similar effects in photon tunneling arrangements has so far remained largely unexplored. Here, we experimentally demonstrate that the tunneling or coupling process in a twisted multicore fiber system can display a chiral geometric phase accumulation, analogous to the Aharonov-Bohm effect. In our experiments, the tunneling geometric phase is manifested through the interference of the corresponding supermodes. Our work provides the first observation of Aharonov-Bohm suppression of tunneling in an optical setting.