The part you're overlooking is these lines:
# Players do a sqrt(X) based on their referee's coin.
The players rotate their qubits conditioned on what the local referee said. This is equivalent to changing the measurement basis conditioned on what the local referee said.
The only relevant property is that the effective measurement bases start off offset by -45 degrees (mostly agree), so that a 90 degree rotation by one player gets you to +45 degrees (still mostly agree) while a rotation by both players gets you to +135 degrees (mostly disagree). The example sets up such a situation.