The relevant FAR relating to flying in the US with a "based on" licence is
14CFR 61.75, which is clear:
Originally Posted by 14CFR 61.75
(b)
(4) Holds a medical certificate issued under
part 67 of this chapter or a medical license issued by the country that issued the
person's foreign pilot license
A Part 67 medical certificate is of course an FAA medical.
So (in the US)
one of an FAA medical or an EASA medical is sufficient.