2Donkeys has it correct as easily verified by actually reading the CFRs.
To reiterate:
If you have an ICAO IR you only need pass the FAA IFP knowledge test to get the IR on your FAA piggyback certificate when first issued.
The medical can be either FAA or from the country that issued the licence on which your piggyback licence is based.
Chickenhouse:
Now we come to penny-pickin'. The 61.75 license may be valid on the basis of a 3rd class medical - if ever anybody issues that on such constellation -, but as the underlying foreign license has to be valid to execute the rights on the 61.75 and I never heard of any foreign license granted execution rights on the basis of the non-ICAO US 3rd class medical, you need the appropriate medical of the underlying license to render that valid.
The CFRs state the 61.75 certificate is valid based on either the FAA medical
or the foreign medical. There is no "penny pinching". The regulation is clear enough and 2Donkeys already quoted it.
Also, the FAA medical is not "non-ICAO". It is ICAO compliant. ICAO does not issue medicals they only publish Standards and Recommended Practices. Filing a difference to a SARP with ICAO does not make something non-ICAO, merely non-standard.