mui/base-ui — reverse-engineered prompt

Build me a polished unstyled React UI library for accessible web apps and design systems. I want headless components that give people the behavior and accessibility out of the box, while letting them fully control the look and styling. Focus on things like strong keyboard support, focus management, screen reader friendly behavior, and clean predictable APIs so the components feel solid in real products.

Please make it feel like a serious open source project, not just a demo. Include TypeScript, good test coverage for behavior and accessibility, a docs site with clear usage guides, runnable examples, and a simple playground where interactions can be tried quickly. The examples and docs should be good enough that someone could copy patterns into their own app without confusion.

Keep the developer experience smooth and the structure maintainable for a component library that can grow over time. If you need details on expected patterns, look up the current Base UI docs online and follow established accessible React component conventions.