¿Por qué se descargan los archivos PHP en lugar de ser procesados por Nginx?

He visto el mismo problema, y muchas resoluciones para ello apuntando a añadir el manejador de tipos, pero eso era para Apache.

Estoy usando Arch Linux, Nginx, PHP y PHP-FMP y no puedo averiguar por qué los archivos php se descargan en lugar de ejecutarse.

Aquí está mi nginx.conf:

#user http;
worker_processes  1;

#error_log  logs/error.log;
#error_log  logs/error.log  notice;
#error_log  logs/error.log  info;

#pid        logs/nginx.pid;

events {
    worker_connections  1024;
}

http {
    include       mime.types;
    default_type  application/octet-stream;

    #log_format  main  '$remote_addr - $remote_user [$time_local] "$request" '
    #                  '$status $body_bytes_sent "$http_referer" '
    #                  '"$http_user_agent" "$http_x_forwarded_for"';

    #access_log  logs/access.log  main;

    sendfile        on;
    #tcp_nopush     on;

    #keepalive_timeout  0;
    keepalive_timeout  65;

    #gzip  on;

    server {
        listen       80;
        server_name  localhost;

        #charset koi8-r;

        #access_log  logs/host.access.log  main;

        location / {
            root   html;
            index  index.html index.htm index.php;
        }

        #error_page  404              /404.html;

        # redirect server error pages to the static page /50x.html
        #
        error_page   500 502 503 504  /50x.html;
        location = /50x.html {
            root   html;
        }

        # proxy the PHP scripts to Apache listening on 127.0.0.1:80
        #
        #location ~ \.php$ {
        #    proxy_pass   http://127.0.0.1;
        #}

        # pass the PHP scripts to FastCGI server listening on 127.0.0.1:9000
        #
        location ~ \.php$ {
            try_files $uri =404;
        root           html;
            fastcgi_pass   unix:/var/run/php-fpm/php-fpm.sock;
            fastcgi_index  index.php;
            fastcgi_param  SCRIPT_FILENAME  $document_root$fastcgi_script_name;
            include        fastcgi_params;
        }

        # deny access to .htaccess files, if Apache's document root
        # concurs with nginx's one
        #
        #location ~ /\.ht {
        #    deny  all;
        #}
    }

    # another virtual host using mix of IP-, name-, and port-based configuration
    #
    #server {
    #    listen       8000;
    #    listen       somename:8080;
    #    server_name  somename  alias  another.alias;

    #    location / {
    #        root   html;
    #        index  index.html index.htm;
    #    }
    #}

    # HTTPS server
    #
    #server {
    #    listen       443;
    #    server_name  localhost;

    #    ssl                  on;
    #    ssl_certificate      cert.pem;
    #    ssl_certificate_key  cert.key;

    #    ssl_session_timeout  5m;

    #    ssl_protocols  SSLv2 SSLv3 TLSv1;
    #    ssl_ciphers  HIGH:!aNULL:!MD5;
    #    ssl_prefer_server_ciphers   on;

    #    location / {
    #        root   html;
    #        index  index.html index.htm;
    #    }
    #}

    server {
    listen  80;
    server_name arch;
    access_log  /srv/http/arch/logs/access.log;
    error_log   /srv/http/arch/logs/error.log;
    location / {
        root    /srv/http/arch/httpdocs;
        index   index.html;
    }
    }

}

Editar para añadir el archivo mime.types:

types {
    text/html                             html htm shtml;
    text/css                              css;
    text/xml                              xml;
    image/gif                             gif;
    image/jpeg                            jpeg jpg;
    application/x-javascript              js;
    application/atom+xml                  atom;
    application/rss+xml                   rss;

    text/mathml                           mml;
    text/plain                            txt;
    text/vnd.sun.j2me.app-descriptor      jad;
    text/vnd.wap.wml                      wml;
    text/x-component                      htc;

    image/png                             png;
    image/tiff                            tif tiff;
    image/vnd.wap.wbmp                    wbmp;
    image/x-icon                          ico;
    image/x-jng                           jng;
    image/x-ms-bmp                        bmp;
    image/svg+xml                         svg;

    application/java-archive              jar war ear;
    application/mac-binhex40              hqx;
    application/msword                    doc;
    application/pdf                       pdf;
    application/postscript                ps eps ai;
    application/rtf                       rtf;
    application/vnd.ms-excel              xls;
    application/vnd.ms-powerpoint         ppt;
    application/vnd.wap.wmlc              wmlc;
    application/vnd.google-earth.kml+xml  kml;
    application/vnd.google-earth.kmz      kmz;
    application/x-7z-compressed           7z;
    application/x-cocoa                   cco;
    application/x-java-archive-diff       jardiff;
    application/x-java-jnlp-file          jnlp;
    application/x-makeself                run;
    application/x-perl                    pl pm;
    application/x-pilot                   prc pdb;
    application/x-rar-compressed          rar;
    application/x-redhat-package-manager  rpm;
    application/x-sea                     sea;
    application/x-shockwave-flash         swf;
    application/x-stuffit                 sit;
    application/x-tcl                     tcl tk;
    application/x-x509-ca-cert            der pem crt;
    application/x-xpinstall               xpi;
    application/xhtml+xml                 xhtml;
    application/zip                       zip;

    application/octet-stream              bin exe dll;
    application/octet-stream              deb;
    application/octet-stream              dmg;
    application/octet-stream              eot;
    application/octet-stream              iso img;
    application/octet-stream              msi msp msm;

    audio/midi                            mid midi kar;
    audio/mpeg                            mp3;
    audio/ogg                             ogg;
    audio/x-realaudio                     ra;

    video/3gpp                            3gpp 3gp;
    video/mpeg                            mpeg mpg;
    video/quicktime                       mov;
    video/x-flv                           flv;
    video/x-mng                           mng;
    video/x-ms-asf                        asx asf;
    video/x-ms-wmv                        wmv;
    video/x-msvideo                       avi;
}