Immigration
A former common-law marriage does not establish a basis for naturalization, as the law requires custody at the time of legal separation, and the end of the marriage was not legally recognized.
http://media.ca1.uscourts.gov/pdf.opinions/14-1858P-01A.pdf
Since 2011. An occasional commonplacing of the U.S. Circuit Courts of Appeal.
Neither an authoritative statement of the law nor legal advice. Quick work.