Frankly, I was baffled as to how this is possible. One side of the relay coil goes to +12V and the other side typically must be connected to ground to complete the circuit and activate the relay. Knowing that the coolant temp switch (F18 in the wiring diagram) was open, the only explanation is that the fan control module (FCM) or possibly the A/C control module was sinking enough current to turn relay on. Wacky sh!t. I don't know if that's a bug or a "feature" but I didn't like it.
The apparent solution was to install a diode in that brown/green wire between the FCM and the tap for the EFK relay. This stopped the EFK fan from coming on with the A/C. I simulated the coolant temp switch activating by jumpering that wire to ground and sure enough (when the engine is running) the aux fans come on at low speed. Wanted to make sure the diode didn't interfere with the FCM and A/C controller getting the right "signal". Seems to work exactly as desired.
Here's a picture of where I put the diode and the wiring diagram from the Bentley manual that I marked up with where the diode goes (note the green circle).


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