Look, if you're after the real meat of a crossover—where Harry actually talks to Annabeth like people instead of plot devices—you can't just search 'Harry Potter Percy Jackson crossover' and hope. The tag system on Ao3 is your strongest weapon. Filter for the 'Percy Jackson and the Olympians - Rick Riordan' and 'Harry Potter - J. K. Rowling' fandoms together, then sort by kudos. But the trick is adding '&' in the relationship tags, not '/'. A tag like 'Harry Potter & Annabeth Chase' means the fic is focused on their friendship or rivalry, not romance, which often leads to more substantive dialogue.
I got burned so many times by fics where Harry just shows up at Camp Half-Blood and everyone immediately trusts him. Lazy. The ones that nail character voice usually have a specific hook, like 'Harry Potter and the Lightning Thief' by lightningfury on FanFiction.net. It's older, but Harry's sarcasm clashing with Percy's is written with a real understanding of their respective burdens. Another is 'Perseus Potter' by the same author, though it's more of a fusion. The interactions there feel earned because the writer spends time on how their magic systems would conflict.
Honestly, I've had better luck on SpaceBattles or Sufficient Velocity forums for this specific niche. The culture there leans towards 'rational' fics and detailed world-building discussions. Authors get grilled in the comments if a character acts OOC, so the resulting stories often have very tight, motivated dialogues. You have to wade through some quests and snippets, but the payoff for genuine character debate is higher.
My bookmark folder for this crossover is embarrassingly large, and the common thread in the good ones is a slow integration. They don't meet and become best friends in chapter two. They argue about methodology, they misinterpret each other's cultural references, and the tension comes from their vastly different approaches to heroism. That's the strong interaction you want.