There's no relay for the compressor, the compressor solenoid is driven by the PCM. If the ground side measures a voltage, then the ground connection at G110 could be bad.
The wires can corrode under the eyelet crimp and insulation where you can't see it, the fix is to cut off the old eyelet, strip the wires back, and add a new eyelet.
If you have an ohm meter try measuring the resistance from the connector ground to the chassis. It shouldn't be more than a few ohms.
More info on the grounds here:
G110 G111 & G112 G111 G112
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