Is your feature request related to a problem? Please describe.
OpenUI supports chat persistence via threadApiUrl and custom hooks (fetchThreadList, createThread, updateThread, deleteThread, loadThread) on ChatProvider, but there is no concrete example showing how to wire this to a real database. Developers need to figure out the schema, API routes, and integration themselves.

Describe the solution you'd like
Add a standalone example app in the examples/ folder (e.g., examples/supabase-chat) that demonstrates full chat persistence with Supabase. The example should include:

• Supabase schema (tables for threads and messages) with migration files.
• Supabase auth integration for per-user thread ownership.
• API routes that implement the persistence hooks (create, load, update, delete threads).
• Real-time subscription support for live updates using Supabase Realtime.
• A working Next.js app wiring it all together with ChatProvider.
• Clear setup instructions (Supabase project creation, env vars, running migrations).

Describe alternatives you've considered

• A generic persistence guide in the docs without runnable code, but a working example is far more practical.
• Using a different database (e.g., Postgres with Prisma, SQLite), but Supabase is popular in the AI/LLM community and offers auth, real-time, and storage out of the box.

Additional context
The example should follow the same structure as examples/openui-chat. Persistence docs already exist at docs/content/docs/chat/persistence.mdx — the example should complement those docs with a production-ready reference implementation.