You have a current loop.
Ideally a pH meter would have a differential input where both the pH and Reference inputs are isolated from ground. This is not the case with cheaper models. They often have a mere 10k resistance to ground or somethimes even less.
Introducing a current into the measuring loop will totally destroy the ability of the pH meter to make a useful reading.
Sometimes it's not even the reference electrode but the communications port (RS-232) which shares a ground with the reference electrode channel. I've had applications where everything worked fine until the RS-232 cable was connected! Luckily enough, there are a few RS-232 isolators on the market.....
Try finding an opto-coupler to place between the electrode and pH meter to counter-act this effect. Actually, you'll need 17 opto-couplers. I remember that German Metrohm had such a device on the market which worked well (at least 4 years ago when I still had contact with them). I just did a Google search and came up empty-handed. I know they're pretty rare and they do cost a few $$.
This is the problem with pH meters of less-than-optimal quality.
Also, grouping the reference channel of 6 different electrodes together is a bad idea. Each reference electrode will have a different potential (can be up to 0.1V) and given the low resistance between them, this will eat up your electrodes. It's just a matter of time. Why not just connect up ONE of the reference electrodes. What type of sample do you have, how far apart are the electrodes?
In case you're wondering, I developed pH electrodes for a while, so I'm pretty much up to speed on the pitfalls of different pH measurements.....
Shane.