Now that a U.S. appeals court has declined to strike down Utah's bigamy laws, it's reasonable to ask: What does the Constitution, properly interpreted, have to say about the topic?
Legally speaking, the issue can be split in two. The first question is whether a state may criminalize marriage to more than one person. The second is whether, in light of the U.S. Supreme Court decision last year to require states to recognize same-sex marriage, there now exists a fundamental right to marry more than one person — and to make states treat plural marriages on equal terms with marriages between two people.
The first one is easier. Under current laws in many states, if you're already married, then it's a crime to marry another person as well. These laws are part of our legal tradition, and perhaps make some sense if you restrict them to bigamists who marry a second spouse without telling them about the existence of the first.
But consider the consolidation of constitutional rights that already exist: I can have sex with any consenting adult under the court's 2003 precedent of Lawrence vs. Texas, which struck down anti-sodomy laws. I can freely engage in any religious ritual under the free-exercise clause of the First Amendment. And using my freedom of speech, I can talk about both my sexual relationships and religious rituals that I've used to solemnize them.
Given these rights, it seems strange that the law prohibits me from forging sexual relationships with multiple partners and calling them my spouses after we've made a mutual religious commitment. If I called them girlfriends or boyfriends, I'm protected by the Constitution. So, it seems indefensible that I can't call them wives or husbands.
It might just be plausible to say that I can't call my life partners legal wives or husbands so long as the law doesn't recognize plural marriages. But even a rule prohibiting me from doing so on the grounds that it might mislead others is almost certainly unconstitutional. In U.S. vs. Alvarez in 2012, the court struck down a law that criminalized lying about winning a medal of honor. The right to lie is therefore enshrined in the constitutional pantheon. If I can lie about a medal, why not about marriage?
The harder question is whether the state should be obliged to recognize plural marriage and treat polygamists equally with those who marry one person. In the Supreme Court's gay-marriage decision, Obergefell vs. Hodges, Justice Anthony Kennedy said that there was a fundamental right to marry the person of your choice and that everyone is owed the opportunity for the equal dignity of marriage regardless of sex or sexual orientation. But he did not say those rights could be extended to polygamy and polygamists.
Logically, however, that extension is warranted unless the government has a compelling interest in preventing plural marriage. Start with the fundamental right to choose a partner. Suppose I am not married and want to choose someone who is already married. My autonomy demands that I be free to make that choice, much as I should be able to choose a partner regardless of that person's sex.