In the rubble of the World Trade Center, integrity monitors shined. A block away, they were unable to stop the corner-cutting that led to two firefighters’ deaths.

Integrity monitors will likely play a critical role in pending city projects around coastal protections and affordable housing. Who will make sure they play it right?

Since scandals of the 1980s, many New York City construction sites have had integrity monitors on hand to police corruption and maintain safety. The system, credited with successes and failures,…