Note that the law criminalizes polygamy, rather than just refusing to recognize polygamous marriages. Here’s the summary from Prof. Howard Friedman (Religion Clause):
In Canada, [a British Columbia trial court] today upheld most of Canada’s anti-polygamy law (Sec. 293 of Criminal Code of Canada) against challenges to it brought under the Canadian Charter of Rights and Freedoms. The suit was brought as a reference case by the province’s attorney general after unsuccessful attempts to prosecute leaders of two FLDS factions....
In today’s decision, Reference re: Section 293 of the Criminal Code of Canada, (B.C. Sup. Ct., Nov. 23, 2011), [the trial judge] concluded that Section 293, while generally valid, is overbroad with respect to its application to children between the ages of 12 and 17.... [EV notes: This means that those children can't be prosecuted for violating the law when they have entered into polygamous marriages -- people who enter into polygamous marriages with them can still be prosecuted.]
The court also found that the statute violates the religious liberty of fundamentalist Mormons, some Muslims and Wiccans — as protected by Sec. 2 of the Charter — but that this infringement is justified by Sec. 1 of the Charter that allows “reasonable limits prescribed by law as can be demonstrably justified in a free and democratic society.” The court rejected arguments that the anti-polygamy law violates various other provisions of the Charter, such as protections of expression and assurances of equal protection. The opinion — which runs 1367 numbered paragraphs in length — includes an extensive survey of the history of polygamy and the alleged harms caused by the practice.
U.S. courts have likewise concluded that laws criminalizing polygamy are constitutional; I discuss that in this earlier post, and cite to two recent cases that have discussed the issue. I think it’s not clear whether, given Lawrence v. Texas, a state may outlaw plural religious marriages where the parties don’t try to make any legal claims based on those religious ceremonies; for more on that, see the majority and the dissent in State v. Holm (Utah 2006). But in any event, so far U.S. courts, like the Canadian court, have not accepted any such constitutional argument.