One possibility in this fanfiction is that Carter and Percy meet in a neutral location, like a modern - day city. Percy is used to the water - related aspects of his powers, while Carter has his unique Egyptian spells. At first, they might misunderstand each other's intentions. However, when some minor mythological creatures start causing trouble in the city, they are forced to cooperate. Percy uses his sword and water - based powers to hold off the creatures, while Carter uses his magic to banish them back to their proper realms. Through this experience, they gain respect for each other and begin to explore how the Egyptian and Greek mythologies are more connected than they thought.
Well, they might start by comparing their powers. Carter, with his Egyptian magic, and Percy, with his Greek demigod abilities. Maybe they'd have a friendly sparring match to see whose powers are more effective. They could also team up to fight a common enemy that threatens both the Egyptian and Greek mythological worlds.
In the fanfiction, Carter Kane and Percy Jackson could initially be suspicious of each other. After all, they come from different mythological backgrounds. But as they get to know each other, they realize they have a lot in common. For example, they both have a strong sense of duty towards their respective worlds. They could exchange stories about their adventures and learn from each other's experiences. Eventually, they might form a strong friendship and work together to protect all of the mythological realms from a greater threat that neither could face alone.