I've always known this, and I'm sure most of you do too, but we never really talk about it. Every smartphone or other device with mobile communications capability (e.g. 3G or LTE) actually runs not one, but two operating systems. Aside from the operating system that we as end-users see (Android, iOS, PalmOS), it also runs a small operating system that manages everything related to radio. Since this functionality is highly timing-dependent, a real-time operating system is required.
This operating system is stored in firmware, and runs on the baseband processor. As far as I know, this baseband RTOS is always entirely proprietary. For instance, the RTOS inside Qualcomm baseband processors (in this specific case, the MSM6280) is called AMSS, built upon their own proprietary REX kernel, and is made up of 69 concurrent tasks, handling everything from USB to GPS. It runs on an ARMv5 processor.