You have at least one fault on the left digital side of your DVM. The 3456A, in contrast to the 3457A, that only displays an undefined "failed" message, boots fine ("3456A"; "AddXXX 0"; "------ ") if the analog boards are disconeced (4 pin connector on the bottom CPU board) or inexistent. It just has a blank "------" display and gives an -3.0000 error during self test.
So if you already have replaced the ROMs and if there isn't any excessive ripple on the +5V rail (big blue cap, IIRC), the next probable cause in my opinion are the RAM chips, especially if those gold capped MOSTEK 4118 are used. Apparently they were made before the problem of gold/aluminium intermetallic was known.
http://en.wikipedia.org/wiki/Gold-aluminium_intermetallic One of those chips also failed in a 3456A that I got. Out of curiosity I removed the lid with a heat gun and analysed the chip under the microscope, where I discovered the gold wire bonding onto aluminium pads. Now after 35 years, solid state diffusion had enough time to happen and AuAl intermetallic formed between the bonding wire and the pad and vibrations, e.g. during shipping caused some cracks that killed the chip.
I still have it, so in case I will get a good microscope with microposition probes once, I might measure the resistance between the pads and the chip leads and see whether I was right.
P.S. I live near you, so in case you get pissed off too hard with it, I can give you (in exchange for the bad boards and a really big box of beer or 50Fr) a known good digital section of the 3456A.