Anomaly that fits your criteria would be the compressor stall situation on a turbo jet.
It is feasible to maintain the engine at min thrust and use as a backup in the event that it is not a lucky day and you lose another engine. Alternatively it could be shut down and then restarted if required. All ways good to leave your options open
But the AOM/FCOM/etc will also mention a caution that continued operation of an engine that exhibits stall tendencies must be done with extreme caution. If high EGT becomes evident or a rapid EGT rise occurs during slow throttle advance or an increase in vibration is noticed than these would require the engine to be shut down