I just finished this article, in which the author discusses the devolution of western society (though I don't think he knew that's what he was writing about).
Courts in the Western World are overrun, and have been for decades. To combat this, legislatures in the West came up with something called Alternative Dispute Resolution ("ADR"). ADR works by litigants agreeing to have their dispute heard, and decided, by an arbitrator. The arbitrator's decision is binding on the parties, and can be enforced by a court.
In Britain, the mother of our judicial system, Muslims have taken advantage of the applicable ADR statute by setting up a number of Sharia courts that act under the auspices of ADR. Up until last year, the rulings of the Sharia courts were nonbinding in nature, but after a clever Muslim identified the ADR rules in Britain, the Sharia courts are now making rulings that are enforceable by British courts.
What kinds of cases are the Sharia courts handling? Well, divorce, family violence, disputes among neighbors, and inheritance disputes.
So, what's the problem? Excellent question.
Islam is not merely patriarchal, but it is anti-woman. Wives are treated as chattel, and can even be beaten under certain circumstances. To have a "family violence" dispute heard by Muslim courts is obnoxious. Consider the following from the linked article:
"In the six cases of domestic violence, Siddiqi said the judges ordered the husbands to take anger management classes and mentoring from community elders. There was no further punishment.
In each case, the women subsequently withdrew the complaints they had lodged with the police and the police stopped their investigations."
So a woman gets beaten by her husband, she calls the cops, a religious court says that hubby needs to go to class, and the woman tells the cops to nevermind. Blackstone must be rolling over in his grave.
The Muslims would cite that submitting to ADR is voluntary. But the root of voluntary is volition, meaning the person making the choice is free to choose either way, uninfluenced by another. In this case, voluntary would mean uninfluenced by the wife-beating husband and the larger Muslim community that supports such activity. With all that pressure, and dwelling with a wife-beater, I can't imagine how a woman in that situation can be said to submit to arbitration of her own volition.
The article also cites an inheritance case where the Sharia court awarded male children double the property of female children according to Sharia law, whereas under British law the children would have shared equally.
Historically, there were good reasons for having such inheritance rules, but with the decline of agriarianism those reasons are essentially gone. In any case, what exists is a situation where the Koran and the writings of Muslim clerics (which is what Sharia is based on) are replacing Anglo-jurisprudence.
By the way, as a funny aside, you saw what types of cases the Sharia courts heard, the article notes that Jews have been using religious courts to settle certain disputes under ADR rules for 100 years.
What kinds of cases are they using it for? "Jewish Beth Din courts operate under the same provision in the Arbitration Act and resolve civil cases, ranging from divorce to business disputes."
Britain has no written constitution. The U.S. Constitution is not perfect, but the Establishment and Free Exercise Clauses are pure genius, and necessary for a free people to exist. I think the Establishment Clause would act as a firewall, and prevent court-enforceable arbitrations based on religious law, but I'm not certain. (Something tells me that if it was Catholic Canon Law it would be struck down instantly, but an "enlightened" Supreme Court may tolerate the existence of Sharia arbitrators.)