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Does the 777-200 do wing to wing automatic fuel balance/transfer?
No, it's done manually. There is a fuel imbalance alarm however.
There is no wing to wing balance system, either automatic or manual.
Balancing the fuel is done via using one tank to feed both engines until fuel in wingtanks is equal. Then it is wingtank to engine again. You cannot transfer fuel from one wingtank to the other.