The DAC probably has a mute when it loses sync. Try Toslink?
FWIW - Ground Loops are a common problem particularly as most audio gear uses ground for the signal (RCA) and many power supplies are leaky to ground - resulting in an imbalance in the signal wires with respect to a ground/shield signal. Even a modest resistance in a shield can induce signals in your interconnects with mere microamps of ground loop (the difference in ground between one socket and another in a house).
In your situation I would go all XLR with properly grounded gear (three prong).
FWIW - Ground Loops are a common problem particularly as most audio gear uses ground for the signal (RCA) and many power supplies are leaky to ground - resulting in an imbalance in the signal wires with respect to a ground/shield signal. Even a modest resistance in a shield can induce signals in your interconnects with mere microamps of ground loop (the difference in ground between one socket and another in a house).
In your situation I would go all XLR with properly grounded gear (three prong).