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A split could give an unintended boost to Scottish nationalists.

“There might be a counter-dynamic in Scotland, whereby a majority of Scottish voters would say, ‘Wait a minute. If you English make us collectively as the U.K. leave the EU, we would then opt for Scottish independence so we could rejoin the EU,’ ” said Mr. Klau.

Mark Briggs in London and Charles McPhedran in Berlin contributed to this report.