How to login to a Docker Registry that has an HTTP prefix set?

I have setup my own Docker Registry, but I did not want it on the root URL so when I created the service I used the REGISTRY_HTTP_PREFIX environment variable and set it to /registry/, thus the URL to the registry is https://tools.example.com/registry. This is being proxied by Nginx which has Basic Auth setup on it.

I tested access to the registry using a Browser and I was able to get it to show that there are no repositories by going to http://tools.example.com/registry/v2/_catalog:

enter image description here

This led me to think that it was workoing. However when I try to login to the registry using the Docker command line, I get the Basic Auth challenge but then it fails to login because the URL is incorrect, e.g.

docker login -u russells -p xxxxxxxx https://tools.example.com/registry/
Error response from daemon: login attempt to https://tools.example.com/v2/ failed with status: 404 Not Found

As can be seen from the error, the prefix is not being added properly. SO how can I login to the registry so I can push images. Is there an environment variable or something that I am missing to make the docker login work properly?

Update - 2017-08-12 2253 BST

I Have been playing around with the configuration a bit, but I am still not getting very far.

As requested here are my configuration files.

nginx.conf

user  nginx;
worker_processes  1;

error_log  /var/log/nginx/error.log warn;
pid        /var/run/nginx.pid;

events {
    worker_connections  1024;
}

http {
    include       /etc/nginx/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  /var/log/nginx/access.log  main;
    keepalive_timeout  65;

    upstream docker-registry {
        server registry:5000;
    }

    map $upstream_http_docker_distribution_api_version $docker_distribution_api_version {
        '' 'registry/2.0';
    }

    server {
        listen 15000;
        server_name tools.example.com;

        # disable any limits to avoid HTTP 413 for large image uploads
        client_max_body_size 0;

        # required to avoid HTTP 411
        chunked_transfer_encoding on;

        location /registry/ {

            # Do not allow connections from docker 1.5. and earlier
            # docker pre-1.6.0 did not properly set the user agent on ping, catch "Go *" user agents
            if ($http_user_agent ~ "^(docker\/1\.(3|4|5(?!\.[0-9]-dev))|Go ).*$") {
                return 404;
            }

            auth_basic "Docker Registry";
            auth_basic_user_file /etc/nginx/.htpasswd;

            add_header 'Docker-Distribution-Api-Version' $docker_distribution_api_version always;

            proxy_pass                      http://docker-registry/registry/;
            proxy_set_header    Host        $http_host;
            proxy_set_header    X-Real-IP   $remote_addr;
            proxy_set_header    X-Forwarded-For $proxy_add_x_forwarded_for;
            proxy_set_header    X-Forwarded-Proto $scheme;
            proxy_read_timeout              900;

        }
    }
}

My Docker Registry service is deployed as registry and is running on the default port of 5000. Looking at this now I think I have got things confused. I do not need the registry to answer on the prefix itself, just Nginx.

For example if I leave the location set to / then I can login, but if I change this to /registry/ then I am not able to. I am beginning to think that the two are conflicting each other.

Registry

I have not set a configuration for the Registry other than the one environment variable - REGISTRY_HTTP_PREFIX, which maybe surplus to requirements in this setup.