Just an idea, have you tried using "Object buffer ID" for the mode of the multi shader, then add a compositing tag to every object that has the material applied. Then in the compositing tag, select the "Object Buffer" tag and enable the first one, make it a 1. Then go onto every other object's compositing tag and do the same but increase the number by 1 each time, or just do a mix of 1, 2, or 3, depending on how many variations you want. It is a little lame because you have to go through and manually set it up the first time, but then everything is still all within one material, and you can modify which variant is applied to which tile by changing the object buffer ID. I've only used this once for lighting an LED tube so I could control each LED within the material, but I haven't put it through much testing, so it'll be interesting to see if it works.
EDIT: So I misread the original post, and assumed you were using the multi shader, not the UVW randomizer, so you could test the randomizer, but I don't know if it would work the same. But you could accomplish what you are doing with the multi shader instead I would think. (You would just create a different texture, or send your texture through a layer node and add a transform to manually transform your texture, and make a few variations then plug them all into the multi shader)