My company has a setup as follows:
As stated by others, you can't access cookies across domains. However, if you have control of the server code, you can return information in the body, and allow your client to read and store that information per server.
In my case, I'm connecting a single client to multiple servers, maintaining an authenticated connection to each one. I need to know when the session for each one is going to expire, so the authentication service returns the cookie, plus it modifies the body of the response to send the relevant data back, so that I can read that data and set my own cookies.
By doing this, I can manually track what I need. Won't work in every scenario, but might for some like me.