Sorry, it must ground through the housing. The paired wires have a diode in them. That keeps the magnetic clutch from frying the car's electronics when it disengages. It's a big magnetic coil just like the ignition coil and can create a big feedback spice when the field collapses.
Anyway, power to that wire engages the clutch.
Here's the diagramOver on the driver's side, down low by the subframe is a small A/C hose with a sight glass on it. On or near there is a switch that turns off the clutch if the high side pressure is too low. This is necessary because the oil is in the coolant, and with low pressure your compressor will burn out quickly.
Run a wire from the magnetic clutch to one side of this switch.
Run another wire from the other side into the cabin near the glove box.
When you remove the glove box you should see the A/C thermostat box on the left side. It has a bulb that goes into the heater box on a thin tube, like a mechanical temperature gauge. The thermostat is controlled by a cable and cycles the clutch to adjust the temperature of the evaporator. It helps to keep the evaporator core from freezing up.
If the heater wiring is not messed up you should be able to find the gray/black wire in the harness in the engine compartment, otherwise you'll have to wire straight to the A/C thermostat.