Indeed.
In fact, my naturalized citizen wife would have a far easier time establishing her citizenship bona fides than I could as a natural born citizen.
Eons ago, for employment related purposes, I needed to prove my American citizenship. So I sent off to the relevant bureaucracy in the State where I was born, and they promptly sent back a brand new properly filled, stamped, signed, and sealed "Certificate of Live Birth" with my name and vital statistics on it. It's not hard to do.
Unless your name is Barack Hussein Soetoro Odinga Whatever Obama.