I was recently informed about a situation where a woman married a man who happened to already be married. The blushing bride had no idea about her husband's prior (and still existing) marriage to the other woman, and understandably was quite embarrassed by her situation. Perhaps more importantly, however, the woman wanted to know what her legal rights were under these strange facts.
This type of situation is what Texas law refers to as a putative marriage. A putative marriage is a marriage that is entered into in good faith by one of the parties (here the wife given her lack of knowledge about the prior marriage) but that is invalid (in this case because of bigamy). Although the marriage in this circumstance is void (because of bigamy), Texas law protects the innocent spouse by making the marriage "putative." A putative marriage gives the innocent spouse who acted in good faith the same rights as a lawful spouse would have had during the marriage.
Without going into the details as to proving good faith, the legal effects of the putative marriage are quite significant. Texas courts have held that a putative spouse has the rights of a lawful spouse in property acquired during the marriage - i.e. community property. As such, once the putative marriage ends, the putative spouse is entitled to a just and right division of the community property, the right to seek temporary support during a suit to dissolve the marriage, and the right to seek spousal maintenance.
Bottom line, if the putative relationship was entered into in good faith, then the court will treat the dissolution of the relationship very similarly to that of a "normal" divorce.