A married couple divorces, within a short time they reconcile, (in God's eyes the wife would have been wrong, she repented and made it right with God and the church). This all happened many years ago, but they just never formally remarried, and the state in which they live in it is legal for either Common Law (under certain conditions) or you can also file a Declaration of Marriage form. Since in God's eyes she was still married, it was never seen as wrong to be reunited with her husband, but should the couple file the Declaration of Marriage or leave as is since both are legal. Either way in the state they live is legal in eyes of law.
Interested as to your thoughts.
"Now to the married I command, yet not I but the Lord: A wife is not to depart from her husband. But even if she does depart, let her remain unmarried or be reconciled to her husband. And a husband is not to divorce his wife" (I Corinthians 7:10-11).
Very few states recognize common law marriages, and a common law marriage in those few states are generally not accepted in other states. The real problem is that common law marriages don't meet the criteria for a marriage. "Yet she is your companion and your wife by covenant" (Malachi 2:14). Common law marriages are not based on covenants.
I understand that the marriage should not have been ended in divorce, but the fact is that legally it was. That they reconciled is great, but they should restore the record of their covenant as well.