The ionization of liquefied noble gases by radiation is known to be accompanied by fluctuations much larger than predicted by Poisson statistics. We have studied the fluctuations of both scintillation and ionization in liquid xenon and have measured a strong anti-correlation between the two at a microscopic level, with coefficient −0.80 < ρep < −0.60. This provides direct experimental evidence that electron-ion recombination is partially responsible for the anomalously large fluctuations and at the same time allows substantial improvement of calorimetric energy resolution.