This is the reason that states in the early republic favored the traditional family arrangements over patriarchal family arrangements, polygamous or bigamous marriage, and associations of free love.
For instance, if a man dies leaving two wives, a court could take into account their de facto relationship, which is bigamous under common law, when distributing the inheritance.
If there were any, she could not marry again, but if she assumed the care of the children, she would have a right to the property from the bigamous relationship.