"I now believe the strange orb was some kind of Thran security device. Supporting this theory, I can no longer recall where I found it."
—Tocasia, journal entry

Mesmeric Orb

Whenever a permanent becomes untapped, that permanent's controller mills a card.

A step in one direction is two steps away from another.

The Brothers' War Retro Artifacts

Double Masters

Whenever a permanent becomes untapped, that permanent's controller mills a card. (They put the top card of their library into their graveyard.)

Salvat 2005

If permanents become untapped during the untap step, Mesmeric Orb's ability will trigger that many times. However, since no player gets priority during the untap step, those abilities wait to be put on the stack until the upkeep starts. At that time, any "beginning of upkeep" triggers will also trigger. Those abilities and Mesmeric Orb's triggers are put onto the stack at the same time.