For the low-gain free electron laser (FEL), an external field is injected into an undulator alongside an electron bunch. Due to phase slippage (because the light is faster than the electrons) the electron and light field are always in resonance. When the electron loses energy to the light field, the fields' amplitude increases. On the other hand, electrons that gain energy reduce the fields' amplitude. To obtain a net gain for the fields' energy, the electrons are injected off-resonance.
What I am wondering; is the emitted radiation in the end coherent?
For only one electron: sure! But the electrons are part of an electron bunch (no microbunching) with a length much larger than the wave-length of the light field. How does that influence the coherence?