The GHZ Game in Black and White

We show that it is possible for three people to win the GHZ game with just the standard real valued misty state formalism, but each party will need to have two balls instead of one. That is, there will be six balls involved, rather than three. In fact, as a general rule, instead of using complex numbers in a quantum computer you can always add a qubit to "tag" complex numbers. But for proofs of non-locality, like the GHZ game, we cannot have a single universal tagging qubit, because all three parties would need access to it and therefore they could potentially transfer information via it. That's where we need to replace the one ball with three that each only need to be accessed locally.