In 2017, a private investigator masquerading as an adviser to a wealthy Indian businessman blundered trying to dig up dirt on an outspoken Russia critic. An undercover operative unsuccessfully tried to prod a former Canadian judge to disparage Jews in the same year. Last year, agents were exposed engineering a smear effort against financier George Soros.

The would-be secret agents all worked for Black Cube, a private Israeli investigative firm often referred to in press reports as a “private Mossad.”

The...