So here are a few things which are interesting about the waif.
She is only seen by either Jaqen H'ghar or by Arya.
She has a very incredible back story, much like Mercy or Cat of the Canals.
She know's what Arya is doing, all the time. She knows when she lies. She can tell where she is.
She tries to force Arya to leave. She makes her insecure.
So here's my theory, get some tinfoil..
The waif is Arya. Arya is suffering from some sort of bi-polar dissociative identity disorder. A case can be made that when Arya came to the House of Black and White, the water that she drank induced this in her. In a way, The Waif is no one. Her only mission is to kill off Arya, the inner Arya, the one who can never be one of them.
This is why Jaqen H'ghar always wants the Waif to leave the room before he begins talking to Arya. He essentially wants Arya to stop behaving in a dual personality manner. He just wants to talk to Arya. So the Waif leaves. Just when Arya is alone, she appears again.
When the waif asks Jaqen H'ghar, "You PROMISED", it's actually Arya. Arya has finally asked to kill her inner self. Al lot of people have observed that Jaqen H'ghar looks a bit pleased with himself when he permits the Waif.
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