Phaethon's Syrtos is a worldwide mission in Genshin Impact that takes place in Enkanomiya. It takes quite a bit of setup to get started, including finding some Seelies and completing another long world mission. Before attempting this quest, you need the Divine Bridle item.
Obtain this item by completing the Hyperion's Dirge world mission. It's a long quest that requires you to get quite a few seals around Enkanomiya. Follow this guide for some pointers on how to complete the mission.
After obtaining the Divine Bridle, you must find the two special Seelies. You will find them at the Temple of the Night.
You must unlock the special gate by finding the two Seelies.
Another prep work you can do before fully starting the mission is to collect Sango pearls. Sango pearls are material for the character's ascension and you will need to place seven of them around the tombs of Enkanomiya.
Once you've found both Seelies, go inside and talk to NPC Clymene. Climene will mark the seven tombs you need to visit on the minimap, but here are the locations of all the tombs:
Return to Clymene once you are done placing all the Sango pearls in the tombs. Clymene will then open a path, the place of soul guidance. Follow this path.
Clymene therefore requires the Divine Bridle. Since you have finished Hyperion's Dirge, you will be able to complete this mission by delivering the Divine Bridle. He will ask you to place the Divine Bridle in the pool of water next to him.
This is the end of the search. You will receive rewards for your problems.