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It was recently proposed that the dark matter-deficient ultra-diffuse galaxies DF2 and DF4 in the NGC1052 group could be the products of a "bullet dwarf" collision between two gas-rich progenitor galaxies. In this model DF2 and DF4 formed at the same time in the immediate aftermath of the collision, and a strong prediction is that their globular clusters should have nearly identical stellar populations. Here we test this prediction by measuring accurate F606W-F814W colors from deep HST/ACS imaging. We find that the clusters are extremely homogeneous. The mean color difference between the globular clusters in DF2 and DF4 is $-0.003\pm 0.005$ mag and the observed scatter for the combined sample of 18 clusters with $M_V<-8.6$ in both galaxies is $0.015 \pm 0.002$ mag. After accounting for observational uncertainties and stochastic cluster-to-cluster variation in the number of red giants, the remaining scatter is $0.008^{+0.005}_{-0.006}$ mag. Both the color difference and the scatter are an order of magnitude smaller than in other dwarf galaxies, and we infer that the bullet scenario passes an important test that could have falsified it. No other formation models have predicted this extreme uniformity of the globular clusters in the two galaxies. We find that the galaxies themselves are slightly redder than the clusters, consistent with a previously-measured metallicity difference. Numerical simulations have shown that such differences are expected in the bullet scenario, as the galaxies continued to self-enrich after the formation of the globular clusters.