What operating system does your AirPods run?

Sounds like a weird question.
Until you realize you have the equivalent processing power of an iPhone 4 in each ear.🙈

Bluetooth audio SoCs are seldom talked about, but a fascinating field.

AirPods specifically run RTKit, a Real-time Operating System targeting small ARM chips, written mostly in C++.

Small Real-time os(s) are often used in audio devices and peripherals, as the slightest hiccup in scheduling would be immediately (aka audibly) obvious. Timings are extremely tight.

Much of the public knowledge about RTKit comes from the Asahi Linux project.
RTKit firmware is actually bundled into iOS updates; a simple extraction of the firmware will show numerous .im4p files (which are really just packed Mach-O).