You might try out Facebook or Google Plus logins - users are accustomed to these flows and both platforms are quite tight now when it comes to real names.

Another way would be using credit-report sites, or having them submit a scan/photo of their ID card/driver license. I do not know if this is legal in the US, though.

IDology has a service for what you're looking for I believe. It's called KYC or "Know Your Customer". I use it on my site to verify identities before opening bank accounts in their name. For my use, we require SSN, DOB, and address as inputs. It checks credit agency and other public data. If user enters any of this info incorrectly, it switches to OOW "Out of Wallet" method and will prompt user to answer questions only they should know, like "What year did you purchase your Ford Bronco?" If they answer 2 out of the 3 of these questions correctly it assumes they are who they say they are.

Not sure what your use case is; but rather than identifying the PERSON is unique, you can identify that their PHONE NUMBER is unique. Require a text message code be entered on screen. Most people have only 1 cell phone; and if push comes to shove their real name is linked to their cell phone in most cases, so they could be found out via court subpoena.