Dashboard on a subdomain

Is there a way to run dashboard on a subdomain of my primary domain? It will be on the same server. I’ve been fighting with it for a while, and maybe I’m missing something obvious that hopefully someone else can point out.

My reason for wanting to do this is so that my public-accessible site can’t tell that I’m logged in. This may seem an odd request, but it makes sense in my situation. If the admin cookie is set at a different domain (or a subdomain), the public site will display as usual, without delivering a different ‘admin’ version of the site.