I suppose so, but then holding them liable for crimes in the land is a bit silly.

I don't think that's why they mentioned "criminal complaint." I suspect (correct me if I'm mistaken) that it's a necessary step to obtaining a subpoena or other process needed to unveil the domain registrant.

Yeah it reads to me (IANAL) as some "legal blackmail", hey give us this info or we sue you for the registrants crimes.

It's an arm twisting move to force give out the name. Once they comply the will move to the next link of the chain and try to twist that arm. It never ends.

Wonder what happens if the registrant has a fake name and address. I guess the domain gets removed?

I don't see the distinction.

They are saying they will hold them liable if they don't turn over the customer's contact info.