First of all, Coulson doesn't explain the necessity of interrupting her interrogation by explaining (or even alluding to, since they were speaking on an unsecure line) that the New Mexico facility was blown up, or that the Tesseract was stolen -- which of course is SHIELD's priority -- but instead by mentioning Clint. And then it takes her all of half a second to shift gears.
(I think stuff like that, as well as a handful of scenes on the Carrier, are pretty much why I glomped onto this pairing despite the fact that they have hardly any screen time together.)
And then the wording Coulson uses, that "Barton's been compromised," is unusual. In spy terminology - at least everything I've been able to Google about it - a compromised position or facility or agent is one whose secret location has been revealed, leaving them vulnerable. In literal terms that doesn't make any sense here, because it's not like Clint was running an undercover op or even on foreign soil.
(Sidenote: Does Natasha know that he is - or was - in New Mexico? Most fic I've read says yes. In the movie she does ask Coulson where he is, but that could be read as 'where is he now?' or 'where is he that he could have been put in that position?' Personally I'm sticking with the headcanon that when they're not working together they pretty much keep tabs on the other's location. BECAUSE I CAN.)
So anyway, in a way it's an odd choice of words. 'Barton's gone rogue' or 'Barton's been turned' would have been less tactful but just as accurate.
However, one of the many dictionary definitions of 'compromised' is to reveal or expose to an unauthorized person and especially to an enemy... which is pretty much exactly right, isn't it? Clint's brain or body or however you want to say it was exposed (to submit or make accessible to a particular action or influence) to an enemy.
Wordplay is fun!
And then of course you have Natasha's use of the word to Clint later on, which I know has been debated and discussed by more cogent minds than mine. When she tells him she's been compromised, is she referring to Loki knowing as much as he does about her past? In that regard she has been exposed. (to make known : bring to light; to disclose the faults or crimes of) To what he said to her before his escape? (Joss's commentary seems to indicate that she was affected by it, IIRC.)
Or is it something else entirely? After all, her next line - "I got red in my ledger; I'd like to wipe it out" - was originally spoken to Loki in reference to Clint, to her being in his debt. There's definitely an invitation to project, at the very least, that she's referring to him again in this scene, although he wouldn't necessarily know that.
Ahhhh I could mull over this stuff all day.