Multiplying the deviation from mean of two time varying voltages
Without getting into the physics / coherence / phases discussion, I'll just point out that analog multipliers exist and can be pretty fast. (Also see "mixers" in the RF world, but four-quadrant operation and linearity might get compromised in the quest for bandwidth.)

Check out Analog Devices' multipliers, which go up to 2GHz.  

It sounds like you're essentially interested in the DC / near-DC content of the product signal.  Your sensors and multiplier have to handle the "around 1GHz", but the rest of the signal chain after the multiplier can be as slow as the length of the "coherence"/correlation effect you're trying to measure.  (Slower is better from a noise perspective.  Don't forget that even on paper, two uncorrelated random signal sources may and will show a non-zero cross-correlation when you're examining only a finite-length time sample.)

Out of curiosity, where in the physics does the 1GHz number come from?  Mean photon inter-arrival time?