Definition: Synthetic testing in production means running functional test cases with synthetic (made-up) data against live production systems, usually at API level. The ideal is "write once, test anywhere": the same test runs in the test environment and in production. Pass or fail may be judged using production monitors and diagnostics.
So what? It extends familiar automated tests into the real environment. It needs careful test data handling, because real user data must not be changed and synthetic data must not contaminate production data.
Examples: Re-engineering test automation so the same tests run against an enterprise product in the lab and its hosted cloud version. Running tests continuously, not just at deployment, with results shown on a quality scorecard.