I never saw any of the Kodak C41 b&w films cross processed in E6, but the lab I used to work at sometimes cross processed Ilford XP2 in E6, and it took on a green tint somehow.
That was with Kodak chemistry in a big Refrema.

The guy processing the film wasn't too keen to do it, as he said he was worried about contamination. I haven't found any substantiated evidence that this would be the case, but when I cross process film using home kits, I always wait until the end of the kit capacity to run those rolls, just in case.

I hope that helps a little bit.

- Thomas