Non-AZ resident, can we register a fiver in AZ? Should we?

I'm sure there are people that do it, Ive even seen it mentioned on You Tube, but most states require you to establish domicile to register the vehicle. With you being listed as Il resident, it would likely save you some money to change it.
One thing I ran into years ago while in the service was towing a trailer with a different states plate than my truck was questioned at a traffic stop. I believe they automatically suspected theft. The cop accepted it but was skeptical.