"You have come to register, have you?" is natural in British English. It's unnatural in American English, because we don't use this kind of question tag.
Normally, a positive statement requires a negative question tag, but British English also allows for positive statements with positive question tags, in some cases. Here's a page that talks about how they're used (courtesy of Su.Ki. in a comment on a post a few days ago):
Nicole has explained the use of "so." Here are some more examples of this usage:
"I'm a teacher."
"So am I." (meaning: "I am also a teacher" / "I'm a teacher too.")
"My coffee is too hot."
"So is mine." (meaning: "My coffee is also too hot."
"I have a cat, and so does my sister." (meaning: "I have a cat, and my sister has a cat too.")
"I'd like to leave now, and so would my husband." (meaning: "I'd like to leave now, and my husband would also like to leave now.")