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This question was answered by @strime:
Ok, eventually ended up finding the solution. The problem is not related to Nginx / PHP-FPM blocking requests while a long running script is executed. It is due to the fact that the server only accepts a maximum number of simultaneous connections coming from the same browser. If you have ajax requests in your pages, it increases this number of connection. Trying to open the website in multiple tabs is then blocked. I’ve not found a solution yet to solve this problem, but at least I know that the other visitors are not impacted.