Hi Pipe,
The isolation transformer here has a ground node of its frame, the ground pins on incoming 120V, and on the outgoing 120V and 240 V sockets.
Secondary 0-120-240 The 0 V outgoing neutral ( wide blade in USA) ( virtual neutral you mention) is not grounded and not connected to incoming neutral.
I have an aluminum ground bar across the bench and all test equip is connected to it via the banana sockets on their front panels.
Because the house is old, I connected the bench ground bar to the house/street water copper pipe,
and hence to the incoming residential utility switchboard neutral/ground close by.
When using the variac, or directly into a rectifier I use 2 wire, no ground from the isolation transformer, the bridge rectifier has DC filters in (+) side and bridge (-) becomes the "bench common 0V dc",and it can be grounded , and the scope probes can be connected to it.
I would not connect an ethernet cable from a bench computer to a house router.
If it came to that, add a router as relay and connect it to the local computer,or else put a wifi card in the computer.
For your comment about ground currents:
Yes, if you are experimenting with power electronics as I do, or RF, the bench common may not be at the building ground potential over the frequency spectrum
I should add that the description above is for forum discussion of one situation of experimenting with inverters on a DC supply.
It is not suitable for any other use.