The Claude Code action has a property track_progress which could be useful to set to true in order to get realtime updates of its progress.
For some reason it does not appear to generate logs of the conversation as it progresses like it used to in v0.
The issue with track_progress is that any GitHub workflow that triggers on comments might be triggering on each update.
https://github.com/anthropics/claude-code-action/blob/main/docs/usage.md