It's hard to say for sure if it's extremely popular. It might be a niche concept within the Percy Jackson fandom. Some fans might be interested in exploring such unique and somewhat edgy scenarios between the two characters, while others might find it a bit too out - of - the - ordinary and not as appealing.
One possibility could be that Percy ties up Annabeth as part of a prank gone wrong. Maybe he was trying to play a joke on her for some past mischief she caused him, but then things spiral out of control and he has to figure out how to untie her before they get into real trouble.
One possible fanfic could be about Percy and Annabeth going on a quest together, not as family but as partners in a more romantic or friendship - based adventure. They could face challenges that test their relationship outside of the family - like bond they usually have in the original stories.
He could also become angry. Percy has a bit of a temper, and he might lash out at Annabeth, demanding an explanation. He might feel betrayed, especially if he doesn't understand the real reasons behind the break - up. In his anger, he could distance himself from their mutual friends and go on solo quests to clear his head.
Well, I've come across some. In many of these fanfictions, the authors explore their relationship further, often adding new adventures or challenges for them as a couple. It's really interesting to see how different writers interpret their connection. For example, some might focus on how they deal with their relationship in the context of the demigod world, like facing new monsters together while also being in love.
It could be a new demigod who has a grudge against Percy and somehow convinces Annabeth that Percy is a threat, leading to her tying him up. So this new demigod would be the antagonist.
One possibility could be that it's part of a prank gone wrong. Maybe Annabeth was trying to play a practical joke on Percy and tied him up, but then things start to spiral as they face some unexpected danger while he's tied. And they have to figure out how to get Percy free and deal with the situation together.
One reason could be Percy's constant danger - attracting monsters and putting Annabeth in harm's way too often. In fanfiction, authors might explore how this constant stress could lead to a breakdown in their relationship.
Perhaps Percy realizes that their relationship has become too codependent. He might feel like he's losing his own identity within the relationship. He could also break up with Annabeth if he discovers some long - hidden secrets about her past that he can't come to terms with, like if there were things she did in the past for survival during a quest that go against his moral code.
One common theme is adventure. Since Percy and Annabeth are from a world full of mythological adventures, the fanfictions often continue this trend. For example, they might go on a new quest together.