I just broke apart an OBiWIFI and the exact model number is RTL8188CUS.
I have just plugged it in to my desktop PC and installed the driver: Realtek RTL8188CUS Wireless LAN 802.11n USB Slim Solo
It's working just fine on my pc but I had to manually install the driver as it wouldn't automatically detect it even though the driver is already on my PC! 8202 appears to represent Broadcom from the searches I found on the net. I only have the one OBiBT dongle so not going to rip that apart!
The third party WiFi dongle I tried uses: Realtek RTL8188CU Wireless LAN 802.11n USB 2.0 Network Adapter and has the following ID: USB\VID_0BDA&PID_8176 but does not work in the OBi. Maybe it has to be a perfect match?
EDIT: Nope, I just setup the OBiWIFI with the Realtek RTL8188CUS Wireless LAN 802.11n USB 2.0 Network Adapter driver... The exact same one the third party uses and it works just fine on the PC so the OBi2xx series ATA's at least are set to use only devices matching the OBi hardware ID.