Podman [supports][1] `docker-compose` now. Just set `DOCKER_HOST` to the path of your Podman socket after activating the service, and it should just work (unless you use Swarm, which is not supported).
It's worth noting for others that (it appears from a quick read, I haven't actually used this yet), the compromise for gaining the "docker-compose" superpower is that you will have to run a podman service (Daemon). This comes counter to some (not all) of the benefits I mentioned above, but is a necessary compromise if one wants the power of compose style orchestration; that is, that there must be some deamon to manage it.
This is not authoritative, I may be mistaken, but this is my educated guess based on a quick read and my knowledge of Docker et al.
The Podman daemon can run as a user service, so the only advantage that would be lost is not having to run a daemon at all (but I don’t think one can avoid that if Docker API compatibility is needed). Is there something else I’m not considering?
No Daemon at all is what I was alluding to. A smaller memory footprint and lower attack surface are advantages of that, it may not be an issue for many/most but is worth pointing out I hope.
[1]: https://www.redhat.com/sysadmin/podman-docker-compose