Can a being be reborn into more that one body in Tibetan Buddhism? I was reading that both the Dalai Lama and Karmapa are both rebirths (emanations) of the Bodhisattva Avalokiteśvara. Also there are other lamas who are also emanations of Avalokiteśvara. 

So is Avalokiteśvara reborn into more than one body and if so is every being that is Avalokiteśvara the same in some way? Or is that just different Tibetan traditions? Generally is rebirth restricted to one body or can it be split across more than one? I appreciate that this will be different across branches of Buddhism so can I restrict my question to Tibetan Buddhism.
Because Buddhism denies independent existence of substantial "self", it does not posit literal reincarnation, as in someone literally being a come back of e.g. Ananda (Buddha's favorite student).

Tibetan Buddhism does have a notion of Tulku though, which is similar. Tulku is a new person that inherits the life stories and responsibilities of a previous person. This is done through teaching this person to deliberately identify with their predecessor, to learn predecessor's history, and to carry on his or her work.

Because the new person's life is so heavily influenced and inspired by the predecessor, we can say that the new person is an embodiment of the abstract energy of the previous one.

In this sense, multiple people could embody the same predecessor. In fact, it is more often  than not a case with Great teachers, that they have multiple "emanations" (usually, of mind, of speech, and of enlightened activity) -- because a regular limited person can't give justice to all of that greatness, so declaring one Tulku would not be appropriate.