Is this the first time this has ever happened when both units were powered up, and you were listening to the tuner? Or have you never had a reason to power up the DAC before, when listening to your tuner?
It sounds like the DAC is emitting some RFI which is getting into the tuner circuitry. The question is, how is the DAC emitting this RFI; via audio interconnects, or AC cable? Is the RFI energy entering the tuner circuitry by each chassis being so close together?
If I really want low noise FM reception, I have to turn off all LED lighting in my house. Most LED lighting and dimmers emit RFI which reaches into the FM and AM bands.