I attempted the 12 player trophy in the very first game I played using multiple tabs on an iPad using Safari, and unlocked the trophy without any problems. I have since tried to using multiple tabs and I get the “only join once for each browser” message.
So, using the same iPad I used Chrome to join the game and could add 12 players to a game using multiple tabs. When I tried a second time I got the message about only joining once for each browser.
In conclusion, and this would need more testing, you can only try for the 12 player trophy on your first game on an individual web browser.