Q:

Expose private Docker registry inside Kubernetes cluster at path `/registry`

I have a private Docker registry inside a Kubernetes 1.13 cluster exposed by a Kubernetes NGINX ingress controller 0.23.0 and the following Ingress object:
apiVersion: extensions/v1beta1
kind: Ingress
metadata:
  annotations:
    nginx.ingress.kubernetes.io/rewrite-target: /$1
  name: my-name
  namespace: my-namespace
spec:
  rules:
  - host: my-domain
    http:
      paths:
      - backend:
          serviceName: registry
          servicePort: 5000
        path: /?(.*)
  tls:
  - hosts:
    - my-domain
    secretName: my-secret

This allows me to pull an existing image as follows:
docker image pull my-cluster/my-image

I would not now change the setup such that the registry is exposed at my-cluster/registry. How can this be done, and is it even possible? 
Changing path to /registry/?(.*) did not do the trick. docker login my-cluster/registry now produces the following error message and docker image pull my-cluster/registry/my-image also does not work. My current guess as to the root cause is that the registry also also use the prefix registry (as configured in the Ingress) for its internal redirects. If applicable, how can this be configured (preferably in the Ingress too)?
Error response from daemon: login attempt to https://my-cluster/v2/ failed with status: 404 Not Found

A:

I have now concluded that referring to a Docker registry by a URL-like location (such as my-cluster/registry) as opposed to a host-like location (such as my-cluster with an optional port) is not possible.
So I'll reserve a separate IP address and certificate for registry.my-cluster and proceed with that.

