Every state I have lived in (including the state of confusion) has several different paths to the same end. On any given day, at any given branch of your state's DMV, you might get a different answer. For instance there is a lien, like a storage or mechanics lien. This is how most towing/impound yards operate. Then there is a homebuilt car. There may be a surety bond type of process and here we have one that is some sort of affidavit that they do a search and see how long the vehicle has been out of the system and then proceed, at their discretion, with a titling process. Or you can find a "friend" in an easier state to deal with, sell them the car and buy it back when they get the cleared paper work.
One of the easiest I have used is to just have a friend with a towing company impound the car, keep it for the required amount of time and put it on their impound auction where you can buy it back as the high bidder (with the understanding that you only pay them the agreed on amount for the service)