Right, this is nothing to do with Jira, it's your browser and usage of it that is broken.
Each time you swap between Jira systems, the browser is destroying the session from the old Jira by overwriting it with the new one.
Your only option here is to use another browser - e.g. open 8080 in IE8 and 8100 in Chrome. Also, you can use anonymous browsing to "sandbox" the sessions if you're using Firefox or Chrome (not sure about IE)
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