Yes; but the detail is not quite like that.
The circumstances where the pilot (or pilots, in a multi pilot aircraft) will need to have the EASA papers are
- flying in EU airspace, and
- operator based in the EU (the exact wording is in EASA FCL somewhere, and is vague), and
- the country where the operator is based has not applied for the April 2014 derogation
So yes an N-reg pilot flying in EU airspace whose operator is EU based will need to have EASA papers in addition to the FAA ones.
There is no definition of what an EU based operator is.
Domicile, or tax domicile, is a yet different concept...