I have Claude run it in a terminal, not through the API. I just ask Claude to spawn a Codex terminal and send it a prompt, and Claude gets a message back when Codex finishes or gets stuck. I do that through my oss project Spaces (usespaces.dev). The steps are in this skill if you want to copy the pattern: https://github.com/yogesh-dhande/spaces-skills/blob/main/ski...

Without Spaces, tmux gets you most of the way: Claude opens a window running codex, sends the prompt with send-keys, and checks capture-pane to see when it's done.

I have definitely had some success at that in the past, I haven't been leaning on it as hard the last month, but for a while 2 months ago I was doing a lot of prompts like: "loop having Codex CLI and Antigravity CLI (`agy`) review your plans and changes up to 5 times until you all reach concensus".