Disclaimer: I do not own The Avengers or any other works from the Marvel Cinematic Universe. All rights are reserved to Warner Bros., Joss Whedon, and any other entitled parties.
To most of S.H.I.E.L.D. they're simply Agent Barton and Agent Romanoff. And the two prefer it that way.
They have their own code for names.
To Fury, they're just Barton and Romanoff. Two royal pains in the ass.
For Coulson, it gets a bit more complicated.
When Clint's in danger, Coulson has the right to call her Natasha. When Natasha's in danger, Coulson has the right to call him Clint.
To enemies, they refer to each other by their working titles; Agent Romanoff and Agent Barton.
To the rest of the Avengers, it's a giant headache to figure out what's the proper name to call them by. It varies person to person.
Steve and Bruce choose the most straightforward option. The assassins are Agent Romanoff and Agent Barton.
Thor calls them by his own strange versions. Natasha is the Lady Widow, and Clint is the Hawk Man.
Tony is the only one that pushes it. He can get away with calling the archer Legolas. But for Natasha, Tony decides to stick with the safe option of Agent Romanoff.
Between the two assassins, the names change and the code becomes a lot more intricate.
The standard for being in the heat of battle is he will call her Nat, and she will call him Hawkeye.
When they're on the job, he becomes Barton. She becomes Natasha.
When they are alone, it becomes different. She becomes Tasha, he becomes Clint.
No one else could dare call her Tasha and still walk away with all four limbs. That is a privilege reserved for her partner only.
The only people Clint allows to call him by his first name are people he considers family. No one else.
Coulson's the only one that understood this part of the code, and he respected it. Welcomed it, even.
Throughout the whole complicated code, there is really only one rule that should never be crossed.
He will never be called Clinton. That name is a part of his old life, the one he swore off.
She will never be called Natalia. That name is a part of her old life, the one she is trying desperately to escape from.
Somedays Clint wonders if the name code is really worth it. It was never really established; just a pattern they happened to fall into.
So he breaks the code that day by calling her Natasha.
"What did Loki do to you?"
"He didn't - I just ..."
It just kind of slips out. Clint's head's been fucked with enough lately, but Clint knows she needs it, so he puts her before himself.
"Natasha."
It's barely a whisper, but it's there. It's a question, a command, a plea. He needs her to remember who she is and what's going on. They're agents, and their working right now. Feelings ... are for afterwards.
"I've been compromised."
Her voice is no longer weak and quavering. It's stoic and steady. Because if she doesn't hide her words behind the euphemism, the real words won't be steady at all.
Clint doesn't know what possessed him to call her Natasha. Maybe it's a good thing he did. Calling her Natasha reminded her where she was, and who she is. And she couldn't afford to be Tasha in that moment, and admit to him the truth.
She had to become Natasha again. To protect what she had to lose. They had a war to go fight after all; they were working. And Clint realized this. So as close as they were to ... no, not yet.
Natasha could hear it in his voice when Clint whispered her name. It was desperately pleading for her to ignore him, to go ahead and confess.
But Natasha knew he was right, even though he desperately didn't want to be. So she swallowed the unformed words on her tongue and they went back to being agents heading to war.
To most people the name code seemed like a hassle. A complex set of rules set up for no purpose.
But it had a purpose. To help them separate themselves; to keep them focused.
If they forgot who they were, then they were dead.
To them, the name code was a necessary evil.