Caddy: Use client cert in reverse_proxy for authentication

Question

I am setting up a local dev environment in which I am proxying certain requests to a back-end server.

When calling the back-end server directly, I get the below browser prompt for a local certificate for authentication.

How can I do the same thing in a reverse_proxy directive in my Caddyfile (passing the certificate to the back-end)? I have tried tracing the calls in Developer Tools but I don't see how the certificate is included in the request (headers).

I thought I could maybe include a tls directive inside the route where I define the proxying to take place, but Caddy complains, saying

tls directive returned something other than an HTTP route or subroute

    route /abc/def/* {
            tls internal {
                    client_auth {
                            trusted_leaf_cert_file "C:\path\to\local.cer"
                    }
            }                               
            uri replace /abc/def /ghi/jkl
            reverse_proxy https://back-end.service {
                    header_up Authorization "Basic xxxxxxxxxxxxxxxxxx"
            }
    }