The spelling of the pepper and the sauce both vary by region.

We're talking specifically about the region in New Mexico in which I live.

There is often a red sauce option :-) But I wanted to get the Hatch chiles while I had an opportunity to as you don't really see them most other places.

The difference between red and green chile is whether it has been dried, but nobody here calls it sauce (even though it almost always is). Hatch is a town that produces it, but they're technically Anaheim peppers. However, if you walk into any restaurant in New Mexico and say "I'd like the Hatch chili sauce on the side" you're going to get a raised eyebrow. The state question is "red or green?"