Why not use Claude App in GitHub (if your code is in GitHub)?

I kick off a prompt as a GitHub issue, Claude fires away on this issue, provides updates as comments and a PR is created for me at the end for review.

It also notifies me throughout, and I can look at the pipelines to see the thinking behind the action.

Or even Claude's native iOS app can achieve what the OP desires. It shows the full interactive transcript as it goes.

I use Remote Desktop to connect to my mac from iPhone because I also want to test local front-end changes remotely.