If both spouses have kept their marriage a secret and don't live together how can they prove that their marriage is in 'good faith'?

We think so...

By their actions, they can demonstrate that the marriage is in good faith. If they choose to live separately but share in everything as typical structured families do; then there is no real difference in their marriage. They may have decided to live apart and that be the only separation from the marriage however; contribute in every other matter, big or small; however their vows to each other and witnesses does not appear to have severed unlike SOME married partners who do reside together. Consider doing your taxes and responded to married lived apart or filing separately. Do either of those questions presume that a marriage is not in good faith. Perhaps their solace is the time away from each other with modern day technology, they perhaps still converse, discuss and solve problems, and visit each other frequently for for what ever reason, still do not reside together and yet; the good faith marriage appears clear. It is the actions of the parties that determine the marriage and their declaration of same. Thanks

Legal marital status of non-citizen

They will likely have a difficult time proving it to the USCIS as the agency only recognizes marriages between citizens and non-citizens as being valid if they represent traditional unions. The couple must reside together as a married couple for at least three years before the non-citizen is eligible for applying for citizenship, assuming he or she already has obtained permanent resident status.