Probably one of the strangest issues I’ve seen, at least it seemed that way at the time.
2 internal web servers experiencing the symptom, one running 2010 OWA and the other a custom web application on 443. All internal users can hit each page just fine. External users cannot hit the pages and they just receive a timeout. However, if the admin logs into either of the two servers locally or via RDP and then you try again externally, it works and they can hit the web pages. This behavior only happened on 443. Customer was just using a Cisco ASA for their firewall with no web publishing.
Customer was a school district and I was reminded of a former life where I worked for a school district where web filtering was common. We found out that external users could only hit the pages when an Admin was logged into either of the servers; not a regular user. Combined with the below Cisco thread I found when trying to potentially pin this on the ASA it seemed a web filter or Intrusion Detection System was killing our connections.
According to the thread a Filter/IDS on the inside could potentially be issuing resets for web traffic that it did not like. In our case it was the customers “iBoss” content filter that started blocking access after a firmware update. It worked when an Administrator was logged into the web servers because it could filter based on the currently logged on AD account and there were exclusions for the Admins.