Could these moons be tidally locked between them or would they get locked to Earth?
What consequences would the orbital cycle of these two moons have on Earth's surface?
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If the Moons have a different orbital period, their relative angular velocity varies, and thus tidal locking is not possible, because axial rotation must be constant. The only possibility left then is if they share orbit, sitting in each others L4 and L5 Lagrange point (like the Saturn moons Dione and Helene). In that case, if they are always facing the same side to each other, they must also be facing the same side towards the Earth. Then both statements are true. Even that arrangement is questionable, as it requires the third object to have negligible mass.