Having lived in New Mexico for roughly five years (and now residing in Northern Virginia), I can say that I wasn't pleased by Anita's.
I think the best way to test authenticity in a place like this is to try their green chile burritos (chile verde). They did use real chiles. I'm not sure if they used green chiles from Hatch, which is what New Mexico is best known for. The burrito was small and expensive, and it was only filled with pork and a light chile verde sauce (and maybe cheese and rice). I'd much prefer they do away with the cheese and rice, and use potato. The burrito could also be improved by using larger chunks of the pork brisket (which wasn't tender enough). The tortillas are not up to par either.
I walked out feeling unsatisfied, though perhaps Anita's might please someone who was NOT from New Mexico, or maybe someone from the West Coast who enjoys their numerous taquerias.