CAMP LEATHERNECK, Afghanistan—Relations between Iran and the U.S. are poisonous, with one exception: an antidote for snake bites.

In a surprising—and irony-rich—byproduct of the Afghan war, the Pentagon finds itself dependent on a scientific research arm of the Iranian government to treat bites by Oxus cobras, Haly's pit vipers and other snakes peculiar to the battlefields of southwest Asia.

Despite...