Varnish not working with WooCommerce

We are attempting to add Varnish to our Woocommerce setup and are running into some troubles.

On a product page (lets say /product/television), if you press the “add to cart” button, you are now redirected to the cart page, as expected, but the item has not been added to the cart.
However, if you do the same thing while logged into an account, it will work fine. I have posted my varnish.vcl below and if anyone has any insight as to what is wrong please let me know.

    # This is an example VCL file for WordPress/WooCommerce/Varnish.
    #

    # Updated to work with Varnish 4


    # Marker to tell the VCL compiler that this VCL has been adapted to the
    # new 4.0 format.
    vcl 4.0;

    # Default backend definition. Set this to point to your content server.
    backend default {
        .host = "web";
        .port = "80";
        .connect_timeout = 600s;
        .first_byte_timeout = 600s;
        .between_bytes_timeout = 600s;
        .max_connections = 800;
    }

    # Only allow purging from specific IPs
    acl purge {
        "localhost";
    }

    # This function is used when a request is send by a HTTP client (Browser) 
    sub vcl_recv {
        # Normalize the header, remove the port (in case you're testing this on various TCP ports)
        set req.http.Host = regsub(req.http.Host, ":[0-9]+", "");

        # The code below makes sure the AJAX "add to cart" function works
        set req.url = regsub(req.url, "add-to-cart=d+_d+&", "");

        # Allow purging from ACL
        if (req.method == "PURGE") {
            # If not allowed then a error 405 is returned
            if (!client.ip ~ purge) {
                return(synth(405, "This IP is not allowed to send PURGE requests."));
            }   
            # If allowed, do a cache_lookup -> vlc_hit() or vlc_miss()
            return (purge);
        }

        # Post requests will not be cached
        if (req.http.Authorization || req.method == "POST") {
            return (pass);
        }

        # --- WordPress specific configuration

        # Did not cache the RSS feed
        if (req.url ~ "/feed") {
            return (pass);
        }

        # Blitz hack
            if (req.url ~ "/mu-.*") {
                    return (pass);
            }


        # Do not cache the admin and login pages
        if (req.url ~ "/wp-(login|admin)") {
            return (pass);
        }

        # Do not cache the WooCommerce pages

        if (req.url ~ "/(cart|my-account|checkout|addons)") {
            return (pass);
        }

        if (req.url ~ "/?add-to-cart=") {
            return (pass);
        }

        # Remove the "has_js" cookie
        set req.http.Cookie = regsuball(req.http.Cookie, "has_js=[^;]+(; )?", "");

        # Remove any Google Analytics based cookies
        set req.http.Cookie = regsuball(req.http.Cookie, "__utm.=[^;]+(; )?", "");

        # Remove the Quant Capital cookies (added by some plugin, all __qca)
        set req.http.Cookie = regsuball(req.http.Cookie, "__qc.=[^;]+(; )?", "");

        # Remove the wp-settings-1 cookie
        set req.http.Cookie = regsuball(req.http.Cookie, "wp-settings-1=[^;]+(; )?", "");

        # Remove the wp-settings-time-1 cookie
        set req.http.Cookie = regsuball(req.http.Cookie, "wp-settings-time-1=[^;]+(; )?", "");

        # Remove the wp test cookie
        set req.http.Cookie = regsuball(req.http.Cookie, "wordpress_test_cookie=[^;]+(; )?", "");

        # Are there cookies left with only spaces or that are empty?
        if (req.http.cookie ~ "^ *$") {
                unset req.http.cookie;
        }

        # Cache the following files extensions 
        if (req.url ~ ".(css|js|png|gif|jp(e)?g|swf|ico)") {
            unset req.http.cookie;
        }

        # Normalize Accept-Encoding header and compression
        # https://www.varnish-cache.org/docs/3.0/tutorial/vary.html
        if (req.http.Accept-Encoding) {
            # Do no compress compressed files...
            if (req.url ~ ".(jpg|png|gif|gz|tgz|bz2|tbz|mp3|ogg)$") {
                    unset req.http.Accept-Encoding;
            } elsif (req.http.Accept-Encoding ~ "gzip") {
                    set req.http.Accept-Encoding = "gzip";
            } elsif (req.http.Accept-Encoding ~ "deflate") {
                    set req.http.Accept-Encoding = "deflate";
            } else {
                unset req.http.Accept-Encoding;
            }
        }

        # Check the cookies for wordpress-specific items
        if (req.http.Cookie ~ "wordpress_" || req.http.Cookie ~ "comment_") {
            return (pass);
        }
        if (!req.http.cookie) {
            unset req.http.cookie;
        }

        # --- End of WordPress specific configuration

        # Did not cache HTTP authentication and HTTP Cookie
        if (req.http.Authorization || req.http.Cookie) {
            # Not cacheable by default
            return (pass);
        }

        # Cache all others requests
        return (hash);
    }

    sub vcl_pipe {
        return (pipe);
    }

    sub vcl_pass {
        return (fetch);
    }

    # The data on which the hashing will take place
    sub vcl_hash {
        hash_data(req.url);
        if (req.http.host) {
            hash_data(req.http.host);
        } else {
            hash_data(server.ip);
        }

        # If the client supports compression, keep that in a different cache
            if (req.http.Accept-Encoding) {
                hash_data(req.http.Accept-Encoding);
        }

        return (lookup);
    }

    # This function is used when a request is sent by our backend (Nginx server)
    sub vcl_backend_response {
        # Remove some headers we never want to see
        unset beresp.http.Server;
        unset beresp.http.X-Powered-By;

        # For static content strip all backend cookies
        if (bereq.url ~ ".(css|js|png|gif|jp(e?)g)|swf|ico") {
            unset beresp.http.cookie;
        }

        # Only allow cookies to be set if we're in admin area
        if (beresp.http.Set-Cookie && bereq.url !~ "^/wp-(login|admin)") {
                unset beresp.http.Set-Cookie;
            }

        # don't cache response to posted requests or those with basic auth
        if ( bereq.method == "POST" || bereq.http.Authorization ) {
                set beresp.uncacheable = true;
            set beresp.ttl = 120s;
            return (deliver);
            }

            # don't cache search results
        if ( bereq.url ~ "?s=" ){
            set beresp.uncacheable = true;
                    set beresp.ttl = 120s;
                    return (deliver);
        }

        # only cache status ok
        if ( beresp.status != 200 ) {
            set beresp.uncacheable = true;
                    set beresp.ttl = 120s;
                    return (deliver);
        }

        # A TTL of 24h
        set beresp.ttl = 24h;
        # Define the default grace period to serve cached content
        set beresp.grace = 30s;

        return (deliver);
    }

    # The routine when we deliver the HTTP request to the user
    # Last chance to modify headers that are sent to the client
    sub vcl_deliver {
        if (obj.hits > 0) { 
            set resp.http.X-Cache = "cached";
        } else {
            set resp.http.x-Cache = "uncached";
        }

        # Remove some headers: PHP version
        unset resp.http.X-Powered-By;

        # Remove some headers: Apache version & OS
        unset resp.http.Server;

        # Remove some heanders: Varnish
        unset resp.http.Via;
        unset resp.http.X-Varnish;

        return (deliver);
    }

    sub vcl_init {
        return (ok);
    }

    sub vcl_fini {
        return (ok);
    }