Tom, I really find it incorrect that you just say it has nothing to do with Owncloud. Owncloud depends on a webserver (Apache or Nginx), PHP and a DB. Specific configuration of the webserver, PHP or MYSQL is important in order for Owncloud to function properly. So each of those components can break Owncloud functionality. You can’t see them separate and say it has nothing to do with Owncloud. That’s just too easy … And if I could find the answer on Google, I wouldn’t be posting my question here.
Anyway, I’ve found a solution by myself. It seems like a SOCKET connection with PHP-FPM does not work very well with Owncloud + NGINX. I fixed it by configuring a TCP connection instead. Check instructions here