I can think of a dozen fics that are similar to what I'm looking for so if you find the right one I'll consider it a miracle: Phil and Clint have some sort of mental bond -- I don't remember if it's A/B/O or something else, but right after the battle, one of them reaches out to the other and Clint realises that Phil dying. Somehow he's able to hold him there mentally long enough for him to stabilize. I know of others where a bond tells Clint Phil's alive but in this one he actually saves Phil.
As you said, there are probably a dozen fics or more that are similar to this, but our followers found a couple that fit pretty closely. Here’s both of them, and hopefully one of them is the one you were looking for, gwynhefar:
I’ll Follow You by sottovoce81, part 1 of the author’s Traits Of An Empath series, which DOES take place after the BoNY.
The Day Clint Barton Missed His Targetby SylvanWitch, part 6 of the author’s Proving the Exception series, which takes place after a non-BoNY incident.
Please let us know if one of these is the one you were looking for, or if neither is!
Thanks to shellumbo and effectricis for the assist.