This was a router issue with our network. The primary router has a DNS Host record for ourgisportal.domain.com that pointed to the GIS Web Adapter's internal IP, 192.168.1.25. Then, internal LAN clients can use the Portal website just as if they are accessing it from outside the network when they are actually not.
Problem was, that DNS host record did NOT have reverse DNS enabled. As soon as it was enabled, the problem went away. In speaking with ESRI support, I recommended this gets added to the Enterprise config. docs; I've never seen reverse DNS listed as a requirement.