Doodling on reports!
He'll never admit it, but he doesn't mind re-writing all of the mission reports as he tells Barton he does. They aren't very succinct, and they wander around the important details a lot. Phil inevitably enjoys reading them, though: Hawkeye is a very good storyteller.
Phil would turn around and submit them directly to Fury, even if the reports are a bit flowery, but he can't. The frequency of profanity and tense-changing isn't ideal but fairly easy to work around. However, what really kills Agent Barton's written reports are the illustrations.
Every operation, every tiny assignment, Phil doesn't think he's ever seen a single report come back from Barton that doesn't have at least one tiny doodle somewhere on the page.
After he re-writes the reports to be officially filed, he doesn't destroy them like he knows he should. He has a secret file in his desk of the originals; he likes to think of these reports as Barton's story.