Apologies for a long reply, this one is wordy.
My first reaction to this was "Why would you want to do that? Just RSS feed, use a sync'ed second database, something."
Which later on became a morbid, watching a train wreck kind of question "How would I do this?" :)
Note that I still think this is a bad idea but this might be a step in the direction you want.
Try this:
domain1.com is the real blog with the real database.
domain2.com is the duplicate that will contain the same posts, pages, and what not just use a different theme. Oh and domain2.com needs to be an Apache web server.
In domain2.com Apache conf file enable mod_proxy, mod_proxy_http, mod_ext_filter and add these lines to the conf for domain2.com (.htaccess won't/shouldn't work)
ProxyRequests off
ProxyPass / http://domain1.com/
ProxyPassReverse / http://domain2.com/
ExtFilterDefine fixtext mode=output \
cmd="/usr/bin/sed s/domain1.com/domain2.com/g"
SetOutputFilter fixtext
ProxyRequests off 'cause open proxies are evil. The ProxyPass commands are so that requests going to domain2.com will cause the web server to pull the pages from domain1.com.
The ExtFilterDefine and SetOutputFilter are to take the pages that domain1.com feeds the web server and modify the web page content so that all domain1.com references are changed to domain2.com. Little things like URL's, style sheet etc. WordPress generates html code with the real blog references and these need to be changed.
Now to see this in action go to these two URL's
http://wp.dembowski.net/
http://dixie.dembowski.net/index.php?wptheme=Tarski
The first one is the real URL of my wordpress blog, the second one in the fake domain2.com. I am using Boren's terrific theme switcher but it should not be to difficult to modify Boren's plugin so that the referer is checked and sets the cookie.
Yes the second URL is a cheat for now. I'll try to see if I can hack Boren's plugin to check the referer and set the cookie that way.
These two web servers are virtual and on the same box. But the domain2.com does not care at all; the original domain1.com is not modified in anyway.
GREAT GUU DO NOT ATTEMPT TO MANAGE WORDPRESS VIA THE SECOND DOMAIN! I have no idea what the sed command would do to the admin interface output or your selections. Commiting any changes to the database using the second domain would be bad.
The drawback is that I was unable to make it work by adding a fake directory (i.e. domain2.com/somepath/) so the directory structure on the duplicate will have to match the directory structure of the original blog.