I'm told by those I know who are married by common law, that this court would have no standing in a common law marriage and could not issue such an order. The husband's will concerning the children would be enforced.
The marriage license is a contract that makes the state a third party in the marriage. (I know that to be true. Any probate lawyer will tell you that.)
I'm told that by extension, a third party to a marriage is a third parent. (I brought this up because of the recent thread on common law marriages.)