Serve the same chat session in multiple websites

@claude,

My users will invoke the agent with a process, and want to load the chat session in the ui to check on how the agent is doing. When they load the chat session, I would like all of the messages to appear, including the partial message the agent is in the process of streaming, and I want the agent's ag-ui event stream to immediately continue populating the latest and partially complete message.

Give me a few ideas on how I can do this.

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*Imported from `waltmayf/agentcore-amplify-fullstack`#66 — originally filed by @waltmayf on 2026-07-07. Migrated to open-source repo; cross-references updated post-import.*

The issue names no files, tests, or entry points. Start by mapping how the process-owned chat session and ag-ui event stream are currently exposed, then document the options for reconnecting a UI to an in-progress session across websites. Done should be a concrete, agreed approach rather than a list of ideas.

Written by the indexing model from the issue text.