Say I have an Illusionary Mask. (Required reading, I know.)
Back? Ok now let's say I use it to play a creature face-down. Doesn't matter - how about Runeclaw Bear.
Then, I use it to play Vesuvan Shapeshifter.
One turn later, I go to attacks. Mask replaces becoming tapped with (TurnFaceUp then become tapped), and I'm attacking with the Bear *and* the VS.
As VS turns face-up, I'm going to copy the bear. Surely I can do this, as that creature card is on the battlefield already, and surely VS will still gain its "turn face down" ability. The question is, will VS be a bear with that ability, or a nameless colorless 2/2 with that ability?