They called her fauxlivia becuase she dressed up and spied on the blue team pretending to be their Oliva.
(Has this all been a huge Halo joke? Red vs Blue?)
They know everyone is real.
When she came back as a friend
, Walter was annoyed with himself that he liked her a little more than the other Oliva.
However I'd just like to take a minute to remind you all of a Bishopian word that really warms my cockles.