His speech was delivered on "Fairness in the Courts". In it, Lord Neuberger suggests what judges might do to ensure that the court system has the confidence of the public. One of these things is to "show... respect to everybody equally".
"That requires some understanding as to how people from different cultural, social, religious or other backgrounds think and behave and how they expect others to behave. Well known examples include... how some women find it inappropriate to appear in public with their face uncovered."
That's been interpreted as indicating support for Muslim women to keep their faces covered in the public setting of a courtroom.
But clearly the speech doesn't actually say that, and it can't change the law. Neither the speech nor the follow-up statement explains whether Lord Neuberger has a view either way.
What does the court system usually say?
Guidance for judges published in 2007 described the veil, or niqab, as "an important element of... religious and cultural identity" for women who wear it.
It said that "for the judicial system the starting point should be respect for the choice made, and for each woman to decide on the extent and nature of the dress she adopts".
A tribunal decision in 2014 noted that "in an increasingly multi-ethnic and culturally diverse society... issues concerning attire and symbols motivated by religious belief and conviction must be handled by all judicial bodies with great tact and sensitivity".
What are the rules on veils in court?
Muslim women don't have an unrestricted right to wear a veil in court (although they are free to in daily life, unlike in other countries).
Judges generally consider that being able to see the faces of the people involved in a case can be important for a fair trial. When someone is giving evidence, in particular, a High Court judgment has described being able to see their facial expression as "essential to assess accuracy and credibility".
There aren't any hard and fast legal rules on how to reconcile these competing interests. More specific guidance for judges was promised in 2013, but doesn't seem to have materialised.
For the time being, judges deal with this tricky situation as they see fit—sometimes using solutions such as "limited screening or minimising the courtroom audience", as judges in a recent case recommended.
Are these compromises legal?
These solutions are backed up by a legal decision, but it's not definitive.
The most concrete statement of the law is the judgment of a Crown Court judge in September 2013. Judge Peter Murphy pointed out that everyone has "freedom of conscience, freedom of religion, and freedom of religious expression. These rights are protected by the Common Law", as well as the European Convention on Human Rights (which is also part of the UK's law).
"If a fair trial is to take place, the jury (and for some limited purposes, the judge) must be able to assess the credibility of the witnesses—to judge how they react to being questioned".
Ultimately, he decided that if wearing a veil "interferes to an unacceptable degree with the Court's ability to conduct a trial which is fair to all parties", it would have to be removed.
His solution was that "the defendant is free to wear the niqaab during trial... [but] if the defendant gives evidence, she must remove the niqaab throughout her evidence". When unveiled, a screen could be set up to shield her from public view—but not from the judge, jury or lawyers taking part.
Is this the last word?
The Crown Court doesn't set binding legal precedents. Judge Murphy expressed the hope in 2013 that Parliament or a higher court (such as Lord Neuberger's) would settle the question. Until that happens, it's likely that case-by-case compromises will continue.
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