I Know What Happened in Budapest.
This is not a made up theory or one-shot. I actually know why Clint remembers Budapest very differently from Natasha. Marvel gave us an answer (although it may not have been what some of you wanted).
In the comic book Secret Avengers (2013-2014) #1, Agent Coulson has a meeting with Hawkeye and Black Widow to ask them if they want to join a team. When they shake his hand at the beginning of the meeting his ring sent “a stream of nanoparticles” that implanted into their skin and traveled through their bloodstreams to their brainstems. S.H.E.I.L.D. then is able to use the remote technology to control their memory implants that are connected to the particles in their brain. (Very important)
Clint and Natasha then go on a mission with Agent Nick Fury in Budapest. During the mission Clint and Fury are chasing after a middleman on a rooftop and when Fury kills the guy after he cooperated Clint gets mad. Fury then tells him that the bullets have a healing factor in them, so not to worry, and Clint responds with ‘Can I get some of those?” and just as Fury is saying “sure” a guy shows up behind them and shots Clint. Clint then lays there as Fury runs after the guy.
Meanwhile, the people they are after comes and picks up Clint to question him but S.H.E.I.L.D. wiped his mind with the particles from the meeting. Natasha then saves him, taking out the villains while Fury kills the main guy they were after, at a separate location.
It then goes to Clint in a hospital bed with Fury and Director Hill looking at him from the outside. They talk about how the “memory implants held up just fine” and then it goes to the scene on the rooftop (the real scene). Everything happens the same except Fury is the one to shoot Clint saying “Sure, you can have mine”. Clint mumbles out “W-Why?!” as Fury runs away, responding “mission objective”
That is why Clint literally remembers Budapest very differently.
