The stars of the US reality television show Sister Wives used the US Supreme Court's recent same-sex marriage ruling to support their case against Utah's polygamy ban, court records show.

The filing with the 10th US Circuit Court of Appeals by Kody Brown and the four women he considers his wives – Meri Brown, Janelle Brown, Christine Brown and Robyn Sullivan – came in response to the Utah Attorney General's appeal of a lower court's ruling in their favor.

Wednesday's filing referred to the Supreme Court's June ruling legalising same-sex marriage nationwide, saying that Utah's position against plural families relies on outdated law.

"This case is about criminalisation of consensual relations and there are 21st century cases rather than 19th century cases," attorney Jonathan Turley said in the 79-page filing. "It is clear that states can no longer use criminal codes to coerce or punish those who choose to live in consensual but unpopular unions."

Polygamy is illegal in all 50 states. But Utah's law is unique in that a person can be found guilty not just for having two legal marriage licenses but also for cohabiting with another adult in a marriage-like relationship when they are already legally married to someone else.

US District Judge Clark Waddoups ruled in 2013 that the cohabitation section of the law violated both the First Amendment's clause on religious rights and the Fourteenth Amendment's due process clause to protect personal liberty. The Attorney General's office later appealed that ruling.

Waddoups also ruled that Utah County Attorney Jeff Buhman violated the constitutional rights of Brown and his women when he investigated them for bigamy.

Representatives for the Attorney General's Office could not be immediately reached for comment on Thursday.

Utah is the headquarters state of The Church of Jesus Christ of Latter-day Saints, or Mormons, which abandoned polygamy in 1890 as Utah was seeking statehood. Some sects and breakaway groups, however, follow the early theological doctrine of plural marriage, thought to bring exaltation in heaven.

The TLC television network first aired Sister Wives in 2010 and it begins a new season next month.

The Brown family and their 17 children, who formerly lived in Lehi, Utah, but now are in Las Vegas, are members of the Apostolic United Brethren, a Utah-based church that follows plural marriage doctrine. In contemporary polygamous marriages, a man is typically legally married to one woman and the others are considered spiritual wives.