Is a photo of my passport on the phone enough, or do I need the original?

You need the original. Art. 19 of the Law "On Police" grants the officer the right to demand an identity document — meaning the physical document itself or an official equivalent (residence permit card, national ID). A photo in your gallery is not legally a document, and in a disputed situation it will not stop the officer from moving to identification under Art. 20 (fingerprints, database check, transport to the station).

In practice: if you don't have the original with you, show what you do have (photo + your Georgian driving licence, a bank card in your name, a residence permit certificate downloaded via my.gov.ge), and ask permission to contact someone who can bring the original. This often works: an officer who sees that you are clearly not "illegal" and that your data can be easily verified against the database will let you go without escalation.

Rule: always carry either your passport or your residence permit card. One of the two. A photo is only a backup.