Dublinpilot has it.
If the state of registry of the aircraft accepts or validates you EASA licence, you can act as pilot in command.
In most countries, the pilot in command is whoever the operator says is the pilot in command. So if your name is in the PIC column of the tech log (or equivalent), you are by definition the pilot in command.
The other pilot would not be able to log this time as anything under EASA rules, or indeed under FAA rules. If he/she does [in Amman, probably he...], this is a bit of a warning sign.
Assuming you can get your licence validated, go for it! Just resist the temptation to have him do all the work... Getting experience in many different countries is better overall than just flying the same flight over and over again.