A chat app could be just a plugin that renders as a text entry window. Type something and hit return to add a new paragraph right above the plugin.
The chat would be centered around the origin server. If the page were forked elsewhere it would become a separate chat there with a common history.
Now we bend the federation rule by allowing visitors to write onto a page that they don't own. A server-side of the chat plugin would handle POST to receive new text and GET for others to discover it.
# Details
One would have to be cautious. Perhaps limit paragraphs to 100 words and pages to 100 paragraphs by discarding the oldest content until within limits.
The client knows when the owner's origin update completes because the journal update returns. It would have to then poll for additional content written in parallel. This is essentially a resync with the server that could revise most of the page.
The visitor's client would be similar but would have to defeat the automatic fork on write, complete the cross-site post (including preflight), and then resync as in above.