I would agree the coaching behaviour before the game, not try to solve it while both of you are already reacting emotionally.
You do not need identical personalities, but the players need one clear environment.
I’d decide three things together: what are we focusing on today, when will we intervene, and what are we deliberately leaving to the players?
If your theme is decision-making, for example, both coaches should know that shouting the next pass is off-limits.
You can also split responsibilities. One coach watches the game, one manages substitutions and player conversations, rather than both shouting competing information.
Afterwards, review one or two moments without blaming each other.
The assistant may not realise they are undoing your message.
Two coaches can disagree about football. That is normal. The key is not making eight-year-olds stand in the middle of the disagreement while the ball is at their feet.
