Do all of your circuits share a common GND? They Should.
Yeah, all grounds tied together. My guess has been that there is some sort of voltage shift happening somewhere along the line that's either corrupting the serial line between the chips, or is causing its reference voltage to determine high/low on that line is getting shifted. Seems possible with the different LEDs going on and off, but at the same time I'd expect the TLCs to already be able to handle that.