If you're looking for a Unix-like, POSIX-compatible, real-time kernel, there's no shortage of projects trying to build one. Ironclad stands out for using the Ada programming language and its formally verifiable SPARK subset.

The Ironclad OS project is writing a new Unix-like OS kernel, aimed at small-footprint and embedded systems, and planning to be realtime-capable. For stronger security, it supports Mandatory Access Control (MAC), which is a big-organization style sort of system, as the US National Institute of Standards and Technology describes.