I normally carry a HK international License when driving anywhere outside Europe/HKG.
Having said that, I’ve never been asked for it when renting in Thailand. The local rental companies normally accept a European license quite happily (in my experience). Make sure you carry copies of all driving/ID documents, and leave the originals in a safe place such as hotel safe etc.
More importantly, make sure that the insurance* on rental cars in Phuket is fully comprehensive and covered by an established agency.
*: Most/all insurance coverage on rental cars in Thailand do not cover any damage to the underside of the vehicle, so don’t be tempted to go off-roading with your little jeep…