I think the tear is a sign that Wells got to Chigurh in their conversation, when Wells called Chigurh crazy.
Chigurh seems to think of himself as a sort of super-human philosopher who sees riddles that other mere mortals can't see - but in reality he's probably just bat-shit insane.
In the scene, Chigurh asks Wells one of Chigurh's trademark wacked-out questions: "If the rule you followed brought you to this, of what use was the rule?" Wells responds by calling Chigurh out: "Do you have any idea how goddamn crazy you are?"
Chigurh doesn't get what Wells is saying at first. Chigurh asks "You mean the nature of this conversation?" It's as if Chigurh has never even thought to ponder his own sanity, despite the fact that he is a fearless and remorseless serial killer.
Wells answers Chigurh with "I mean the nature of you," driving home his original comment. Then Wells says "You can have the money, Anton," using Chigurh's first name, a reminder that Wells is intimately familiar with Chigurh, which lends all the more credibility to his "crazy" assessment.
In the next moment, the phone rings. After thinking for an instant, Chigurh shoots Wells dead. This turns out to be a rare bad decision by Chigurh, as a living Wells would have been able to draw Llewelyn to Chigurh right there in Mexico... instead, Chigurh ends up having to play the Odessa card.
And in the wake of the shooting, before pulling his wits together and talking to Llewelyn, Chigurh seems lost for a moment. From the screenplay:
Aside from his finger on the trigger, Chigurh hasn't moved. He sits
staring at Wells's remains for a beat.
Now his look swings onto the phone. He watches it ring twice more.
Overall, I think the conversation with Wells made Chigurh, if only for a moment, contemplate some unpleasant thoughts about himself - hence the tear.