Carnet for Jordan, Syria, Turkey?
 

I'd like to get up-to-date and if possible, definitive, info as to whether a Carnet du Passage is needed for Jordan, Syria and Turkey (heading from Oman).

I have heard that "no, it isn't", for Jordan and Syria but that alternatives are restrictive/expensive, and I've heard that "yes, it is".

Might or might not be relevant that we'll be in an Oman-registered vehicle and thus with Oman licence plates.

You don't need one. Use the search function and you'll find loads of posts on the same subject.