One common theme is power imbalance. Since Annabeth is bullying Percy, it often shows that she has some sort of power over him, whether it's social status or knowledge. Another theme could be jealousy. Maybe Annabeth is jealous of Percy for some reason and that's why she bullies him.
Percy could stand up to Annabeth. He's a strong character. He might confront her directly about her bullying behavior and tell her it's not okay. This could lead to a turning point in their relationship, where Annabeth realizes she's been wrong.
Yes, there are some. In some fanfictions, after Annabeth bullies Percy, she realizes her mistake. Maybe it's because she sees Percy's good qualities or she herself grows as a character. For example, she could start to protect Percy from other real threats, showing that she has changed and redeemed herself.
She might initially be shocked. Annabeth is a strong character, so she could start by trying to stand up to Percy verbally, asking him why he's being so mean.
There could be some misunderstanding between them. Maybe Percy did something that Annabeth misinterpreted, and instead of talking it out like normal people, she decided to bully him. For example, if Percy accidentally took credit for something that Annabeth thought was hers, it could start this negative behavior.
Often, after Percy protects Annabeth, there's a moment of connection between them. It might be a simple thank - you from Annabeth, but it can also lead to a deeper conversation about their feelings. In some fanfictions, this event is a turning point in their relationship, making them realize how much they care for each other. And sometimes, the bullies come back for revenge, leading to more action and Percy having to protect Annabeth again.
There are quite a few on Archive of Our Own (AO3) as well. Some might be part of a larger series where this scenario is just one of the many interactions between Percy and Annabeth. You can use the tags on AO3 to narrow down your search specifically for the 'Percy protects Annabeth from bullies' theme.
In a large number of Percy and Annabeth fanfictions, they are indeed presented as soulmates. Their shared adventures, the way they understand each other without many words, and the deep loyalty they have for one another all contribute to this perception. Fanfiction authors often build on their existing relationship from the original works and enhance it to the level of soulmates. They might show it through things like Percy being able to sense Annabeth's distress from afar, or Annabeth always knowing exactly what Percy needs in difficult situations. This soulmate - like relationship makes for very engaging and heart - warming stories in the fanfiction realm.
It could be that Annabeth has a misunderstanding about Percy. She might think he's a threat to her position or status at the camp, so she bullies him to try and keep him in check.
It could be Chiron, the centaur. He's always keeping an eye on the campers and would likely be concerned if he caught them up to something. He might give them a stern talking - to and some extra chores as punishment.