Can The Syren (Magma Chamber) move an allied unit to my base?

Ruling: No, you cannot move an ally's unit to your base. The effect should move the unit to its controller's base (the base of whoever currently controls that unit).

Nuances:

• The controller is different from the owner - the controller is whoever currently controls the card, while the owner is whoever brought it to the table
• If a unit has been stolen via effects like Possession or Blind Fury, it goes to the current controller's base, not the original owner's base