If you lose control of Tidebinder Mage between when it entered the battlefield and when the ability resolves, the target creature will become tapped. However, it won’t be affected by the other effect and will untap as normal during its controller’s untap step.

If another player gains control of Tidebinder Mage, the creature will no longer be affected by the ability preventing it from untapping, even if you later regain control of Tidebinder Mage.

If you gain control of the creature that’s being prevented from untapping, that creature won’t untap during your untap step for as long as you control Tidebinder Mage.

The ability can target a tapped creature. If the targeted creature is already tapped when it resolves, that creature just remains tapped and doesn’t untap during its controller’s next untap step.