Oneworld Explorer (xONEx) tickets require both operating aircraft and marketing code to be a oneworld member or affiliate. Codeshares are allowed, but both operating aircraft and code must be oneworld (eg, AA operated QF code is allowed, KE operated JL code is not). Some exceptions apply for QF coded JQ flights.
Global Explorer (xGOLBnn) tickets have more airlines allowed including the Jetstar brands, Fiji Airways, Westjet, Alaska, Aer Lingus, Meridian and Bangkok Air.
QF coded Air Tahiti Nui (TN) flights are allowed on the Global Explorer, but not the oneworld Explorer.
EK, either operated by or as a codeshare, are not allowed on a oneworld product and never have been. If EK was used on a prior trip, then it was either not a oneworld product (such as a QF RTW fare or award) or it was a service recovery airline can brake their own rules change.
The oneworld online tool booked xONEx tickets, TN flights are only allowed as QF coded flights on xGLOBnn tickets, so QF3816 will not show up on the xONEx booking tool.