Nginx PHP-FastCGI: Serving a full PHP project under baseurl/backend/ instead of baseurl/

I'm trying to serve my PHP project under mydomain/backend as I want to reserve my base url for a Node / Vue project. I boiled down the problem to the fact that I am not able to have the PHP app run under /backend due to the interaction between the try+files and the ~.php block.

Any suggestions on how to realize this? Everything works fine if I simply remove /backend and access the app through my root location /

server {
    listen 80;
    listen [::]:80;
    root /var/www/backend/public;
    index index.php index.html index.htm;
    server_name your-domain.com;

    #location ^~ /backend/ {
    location /backend {
        # This is the folder that index.php is in
        try_files $uri $uri/ /index.php?_url=$uri&$args;
    }

    location ~ [^/]\.php(/|$) {
        fastcgi_pass unix:/var/run/php/php7.3-fpm.sock;
        fastcgi_index /index.php;

        include fastcgi_params;
        fastcgi_split_path_info ^(.+?\.php)(/.*)$;
        fastcgi_param PATH_INFO       $fastcgi_path_info;
        fastcgi_param SCRIPT_FILENAME $document_root$fastcgi_script_name;
    }
}

The weird thing is, the relative routing of the PHP project is actually triggered when navigating to /backend, but the actual content of the project is missing. I want the project to be served as if the root of the project was /domain/backend, with all URLs in the project having that /backend prefixed

I tried to add /backend in the ~.php block as well, to no avail.

UPDATE:

Thank you for the response. I have come closer to the solution, but not quite there yet.

Following your tip, and the following link, I've learned the following: How to properly configure alias directive in nginx?

1. try_files should not be used if using alias inside a location block due to the longstanding bug
2. fastcgi_param SCRIPT_FILENAME $request_filename; should be used instead of fastcgi_param SCRIPT_FILENAME $document_root$fastcgi_script_name; if you use alias
3. If you do point 2, you should use index index.php; instead of fastcgi_index /index.php;

However using the updated.conf file, my PHP app reroutes still to the original URL.

What is wrong with the rewrite line? Why does /backend still route back to as if it were the project was in /? It is being fetched by the PHP app, as the Node app goes 404 for all other random non existing sub-URI's.

Do the lines

fastcgi_split_path_info ^(.+?.php)(/.*)$;

and

fastcgi_param PATH_INFO $fastcgi_path_info;

undergo any side effects due to placing them nested in an aliased block?

How do I properly try_files $uri $uri/ /index.php?_url=$uri&$args; to work inside the alias block?

server {
    listen 80;
    listen [::]:80;
    server_name your-domain.com;

    location ^~ /backend {
        alias /var/www/back/public;
        index index.php index.html index.htm;
        if (!-e $request_filename) { rewrite ^ /backend/index.php?_url=$uri&$args last; }

        location ~ \.php$ {
          fastcgi_pass unix:/var/run/php/php7.3-fpm.sock;
          index index.php;
          include fastcgi_params;
          fastcgi_split_path_info ^(.+?\.php)(/.*)$;
          fastcgi_param PATH_INFO        $fastcgi_path_info;
          fastcgi_param SCRIPT_FILENAME  $request_filename;
        }
    }

    location / {
        # node/Vue project
        proxy_pass http://localhost:3000;
        proxy_http_version 1.1;
        proxy_set_header Upgrade $http_upgrade;
        proxy_set_header Connection 'upgrade';
        proxy_set_header Host $host;
        proxy_cache_bypass $http_upgrade;
    }
}