The ruling on this kind of relationship is based on how this secret marriage has been contracted. That is whether it took place in the presence of a guardian and witnesses, and in the form of marriage with dowry or not. If it did take place like that, and nothing was missing from it apart from documentation then it is a legal marriage. If it is done through an agreement between two parties for the sake of sexual enjoyment between them, then it is fornication.
However as for a marriage that fulfills the conditions, it is permissible even if the state does not document it, provided that it is attended by the witness, and that appropriate mechanisms are taken to preserve the rights of the wife and children, especially with the man’s need for polygamy.