The CEO may not have a fiduciary duty towards minority shareholders, but the controlling shareholders do, in this case it sounds like they are one and the same.
If you have to break a tie, then you aren't a controlling shareholder. The idea behind controlling shareholding is that as long as you follow your duties to the minority you can install whoever you want (perhaps yourself) on the board, the board can then install whoever they want as CEO (perhaps yourself)
These things get really murky in private companies where the board, controlling interest, and CEO are shared by one individual.