Globular clusters can form inside their host galaxies at high redshift when gas densities are higher and gas-rich mergers are common. They can also form inside lower-mass galaxies that have since been accreted and tidally disrupted, leaving their globular cluster complement bound to higher-mass halos. We argue that the age-metallicity-specific orbital energy relation in a galaxy's globular cluster system can be used to identify its origin. Gas-rich mergers should produce tightly bound systems in which metal-rich clusters are younger than metal-poor clusters. Globular clusters formed in massive disks and then scattered into a halo should have no relationship between age and specific orbital energy. Accreted globular clusters should produce weakly bound systems in which age and metallicity are correlated with eachother but inversely correlated with specific orbital energy. We use precise relative ages, self-consistent metallicities, and space-based proper motion-informed orbits to show that the Milky Way's metal-poor globular cluster system lies in a plane in age-metallicity-specific orbital energy space. We find that relatively young or metal-poor globular clusters are weakly bound to the Milky Way, while relatively old or metal-rich globular clusters are tightly bound to the Galaxy. While metal-rich globular clusters may be formed either in situ or ex situ, our results suggest that metal-poor clusters are formed outside of the Milky Way in now-disrupted dwarf galaxies. We predict that this relationship between age, metallicity, and specific orbital energy in a L* galaxy's globular cluster system is a natural outcome of galaxy formation in a {Lambda}CDM universe.