Consider two identical gears touching each other. The system is friction less. One has a handle that you use to apply a torque on the entire system. If you turn the handle, there will be a non-zero torque on the system, but since the gears are identical and would have equal angular speeds but rotate in opposite senses their angular momenta would add to zero at any time. So how can the angular momentum of the system remain zero (i.e. never change) when there is an external torque being applied to the system?
Edit: The two gears are fixed on axles. And what if, to simplify, you are just applying a tangential force on the edge of one of the gears (no handles involved) to supply the torque?