Actually I've done the hash on this approach and this still results in a losing game for C. Instead of HH and TT being winners with this approach, HT and TH be

Actually I've done the hash on this approach and this still results in a losing game for C. Instead of HH and TT being winners with this approach, HT and TH become winners with this approach. Any combination of opposite or agreement will result in the same net odds. The way for C to win is for A or B to treat it randomly. A very interesting experiment would be to test how long it would take a random human subject A with no contact with a consistent B (but a memory for past results) to figure out how to win (in cases where they were told winning was possible, impossible, or neither).