CirrusI traced it down to a Cipher issue. 12.0 disables any cipher that supports SSLv3 by default. I had to enable the following cipher to get it to work DHE-RSA-DES-CBC3-SHA.
EmployeeDoes that URL require authentication or cookies ? If not, reduce the complexity of the problem by making a request for it from the commandline of the BigIP, and see what that returns. The -v will show you everything the client (curl) sends, as well as everything the server returns.
curl -vk http:///eumcollector/beacons/browser/v1/eum-aab-aum/adrum
If that doesn't work, direct the request directly toward a pool member,
curl -vk http:///eumcollector/beacons/browser/v1/eum-aab-aum/adrum
Try the pool member directly from your browser, if you routing allows it, or use a browser running on another another pool member if you have that available to you to use.
Look in the /var/log/ltm log file to see if anything appears there when you're accessing the problematic URL - particularly any iRule errors (which will cause a RST to be sent to your client, and the browser to display a blank page)