FAA complex rating counts as differences training?

[Zonkor]

I have an FAA complex rating for ASEL-category airplanes (essentially equivalent to SEP in EASA-land). I also have an EASA PPL (SEP). The UK CAA is my "competent authority".

Question: Can I fly airplanes with constant-speed propeller on my EASA license?

In other words, does the FAA complex rating count as differences training for the CAA, or do I need to do differences training again with an EASA instructor (and get a logbook sign-off)?

Thanks