Is there a unique operator for the phase difference of two quantum fields?

The recent operational approach to the phase operator of a quantized electromagnetic field is applied to the problem of determining the phase difference between two weak coherent fields in two different ways. The phase difference is determined directly, and indirectly by measuring each field in turn against the field of a strong local oscillator. We show by measurements with an eight-port interferometer that these two different schemes lead to quite different outcomes for the probability distribution of the phase difference, in agreement with our theory, but in disagreement with most other theoretical approaches.