Your great-great-grandparents may have decided to live separately without ever divorcing because “we don’t believe in divorce, but can’t live together either.”
In cases like this, there won’t be divorce records, but it is possible that a court action for “separate maintenance” might have been filed. This would have kept the couple “married,” but contain information similar to a divorce.
Couples might also have lived separately without any type of court record or agreement. I had an uncle who lived on the farm while his wife lived in town and an aunt who lived across the street in a separate home from her husband. Her home did not have indoor plumbing–his did. When he would go to a nearby larger town to run errands, she’d go across the street to his home just to use the indoor restroom.
Census records and city directories may hint at these separate living arrangements without providing specifics.