Arduino robot wire follower + how to position on wire
If wire is located in the middle, signals from both coils are in phase. You could measure the phase difference with phase comparator. It would require a reference signal that drives a closed loop PLL. This reference signal goes to phase comparator, the output of phase comparator controls the VCO, then the output of VCO is fed back to phase comparator.
Now when VCO outputs the signal wich is in phase with reference signal you can add 2 phase comparators and compare the phase shift for each coil.
The only problem is where to get the reference signal. You could mount a 3rd coil in the middle, or you can add the signals of two coils if they are the same amplitude, perhaps an additional AGC apmlifieris needed for each coil.