Is it possible to serve static html from php-fpm?

I've dockerised a wordpress application, now I'm being a purist and don't want to include nginx in the docker container and don't want to share any state between the nginx container and php-fpm container.

I want to run php-fpm as a standalone webserver (like unicorn in ruby or gunicorn in python) serving all the content (html, css and images) for the wordpress site. And run a nginx reverse proxy in front off it (caching static content). That way I can keep the separation of concerns and I don't have nginx forwarding traffic to and nginx server.

The default configuration for php-fpm only allows php files to be processed. Can the php-fpm conf that be changed? How? Is it a good idea?

You could, but the performance would be terrible, since everything would go through the PHP interpreter. This would also introduce an obvious, massive security problem.

This is exactly the sort of scenario in which you should be using shared data volumes.