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Update: I installed ShowIP for Firefox as suggested by je44ery and ran through the same steps. (Thankfully, it's quite consistent.)
While browsing the site, the IP showed up as 188.8.131.52 for all pages, including the invisible/blank page. When I changed the extension from .org to .com in the address bar and reloaded the page, the IP remained as 184.108.40.206, and the page displayed just fine. As in the first time, the only difference is that when the page showed this time, I'm not logged in. I checked the cookies (via Tools|Page Info) and found that I have only the 'userpass' cookie on .org and no cookie for .com. Is there any information you'd like me to pull from my browser/OS/... to help? It's consistent, as I've tried the same sequence a dozen times over the last few days and always get the same result. (Note: on the work computer the "invisible" page comes up as a "network error" page as described in the OP.)
Well, I installed Wireshark, but don't really know how to sift through it efficiently, so any tips would be handy. (Or I could install another tool, if that would be better.)
In the morning, I'll restart my machine, flush out the firefox cache and capture a session. Then I'll save the session and start looking for clues.
Note: when I was able to read the content, it was by editing 'www.perlmonks.org' to 'www.perlmonks.com', which brought up the page, but I wasn't logged in. (Does that mean that an old cookie might be the problem? I used to use the .com address by default, but changed to .org when I started having some issues with .com in the time period of the previously-mentioned thread. The thing I find odd is when I bring up a new node and a minute later have the problem bringing up a different node. I would've suspected that the DNS entry would've been the same one.
If I find anything, I'll update the post and message you. If there's anything specific you'd like to see (other than what you just mentioned), let me know.
When your only tool is a hammer, all problems look like your thumb.