Swampwalker, I understand that the buzz only happened after the Joule returned from repair, and I also assume from your answer to Bwhite that the Supratek does not do this with other amps, e.g. the Berning.
If part of a circuitry has a weak ground--for example, the return (ground) connection on an interconnect or RCA socket--then this can cause the signal to "find ground"--i.e., induce a current over dielectrics--in other parts and induce buzzing in tranformers.
Once the Joule and Supratek are connected via interconencts they are essentially a single circuit--and you are getting noise in both power supplies.
My guess--and this is simply a guess--is that either the process of connecting and disconnecting interconnects has created an intermitant short in one of them, or shipping may have caused some minor weakness in one of the many internal ground paths in the Joule. Because every component is different, it is coincidential that connecting the Yamaha C2A "curcuitry" doesn't induce errant current flows, but connecting the Supratek's does.
Try: 1) swapping a few interconnects; 2) inspecting, flexing, and testing them for continuity with a voltmeter; and/or 3) opening the Joule and looking for weak solder joint.