If the marriage certificate was not properly signed and/or filed, the couple are likely not married. However, it is better to consult with a family law attorney, because there may be facts that could affect my answer. Good luck.
Georgia will only recognize a marriage performed in another state (or possession) if it is a true legally performed marriage between a man and a woman. There is no longer any common law marriage recognized in Georgia. Your question then becomes whether or not Puerto Rico recognizes your marriage. You should consult with an attorney there. If there is any doubt, and you both wish to dissolve the marriage, you can consent to an annulment in your Georgia county of residence.
I am exclusively a family law attorney, practicing primarily in the metro Atlanta, Georgia trial courts. However, I handle appeals from anywhere in Georgia.
You mention that the couple "got married in Puerto Rico." Then you mention technical errors regarding signatures, etc... The whole issue comes down to whether they are still considered married under the law of Puerto Rico. If Puerto Rico law still considers them married despite the lack of the signing of the marriage document then a divorce is necessary. If Puerto Rico law does not show them as married and under the circumstances would not consider them married then they would not need a divorce because they were never married.
You need the opinion of a Puerto Rico Matrimonial Lawyer to determine whether or not the couple is considered to be married in Puerto Rico or not.
Law Office of Christopher A. Pearsall, Esquire (401) 632-6976 - Exclusive Divorce and Family Law Atty. Disclaimer: This is NOT legal advice. It is GENERAL INFORMATION ONLY, and does NOT establish an Attorney/Client Relationship with you. My answer cannot address your specific legal situation and you should not rely upon my answer in your legal matter. The GENERAL INFORMATION I provide here is NOT a substitute for legal advice provided by a competent and licensed legal practitioner in the jurisdiction where your case would be heard after the attorney is informed about all the facts and circumstances surrounding your case. You should always obtain professional legal advice and you should never act on any information found on the internet, no matter how reliable it may be or seem.