Exploiting a fundamental characteristic of galaxy assembly in the {\Lambda}CDM paradigm, the observed spatial biasing and kinematics of metal-poor globular star clusters are used to constrain the local reionization epoch around individual galaxies. Selecting three galaxies located in different environments, the first attempt at constraining the environmental propagation of reionization in the local Universe is carried out. The joint constraint from the three galaxies (z_reion = 10.5^{+1.0}_{-0.9}) agrees remarkably well with the latest WMAP constraint on z_reion for a simple instantaneous reionization model. More importantly, the range of z_reion values found here are consistent with the global range of z_reion estimates from other observations. We furthermore find a 1.7{\sigma} indication that reionization completed in low-density environments before the intergalactic medium in high-density environments was reionized. This is consistent with certain theoretical models that predict that reionization was globally prolonged in duration, with neutral hydrogen pockets surviving in high-density environments, even after the surrounding regions were reionized. More generally, this work provides a useful constraint on the formation history of galaxy stellar halos.