For our project, we were able to borrow some sheets of special reflective material, designed to be illuminated by a ring of LEDs that sit around the camera lens. I don't know how much these cost, but I suspect they're quite expensive, so we were lucky to know someone who could let us use it. They pretty much eliminate the need to light the back screen, because it's self-luminescent. We just had to make sure the subjects were well-lit, otherwise it gets a bit contrasty.
We're still using it now. It's been very handy.
I think it's this stuff
, or something very similar to it.
EDIT: This is the one we're using.