Thanks @evenreven, actually Stimulus connect is working well. When a broadcast stream is appended to the DOM, the Stimulus connect() method does fire. (Mostly what has been advised higher in the thread) Though it seems that it fires before the whole stream is downloaded/appended… Then the JS in the connect method() doesn’t do its job properly as some DOM nodes are missing.
events like DOMcontentloaded or tubo:load obviously don’t work. turbo:frame-load does work but only on first page paint, not with the streams.
Then having a hook for a completed stream would be really helpful… Although today I will move the controller outside the stream and play with MutationObserver, there is a chance I can still do what I need.
EDIT: The more I think about, the more I realize it doesn’t probably have to do with Turbo or Stimulus. My problem is probably rather about the paint of the page…