There is a tombstone in the local cemetery. I can't remember the husband's name, but he had two wives, both were named Mathilda. One can only imagine how confusing this might be for his descendants.
Usually a new wife has a different name. If I researched this individual, the age of his wife might change significantly in census records, her birth place may suddenly be different, or other pieces of information may be inconsistent. Keep in mind that if the details on a spouse are different, it might because there was a different spouse--just one with the same first name.