When I was reading books before my daughter was born, I came across an idea that really stuck with me. I don't remember source book unfortunately, possibly Brain Rules for Baby, http://www.brainrules.net/brain-rules-for-baby.

It was addressing this topic of parents arguing – an extreme form of disagreeing – in front of the children. Studies have shown that parents arguing can be traumatic for children. It makes them feel unsafe and emotionally threatened, and it results in increased depression, anxiety, aggression, etc.

However, what most of the studies failed to include in their analysis is what happens after the argument. Children who see their parents constructively resolve their argument, e.g., through negotiation and compromise, actually show decreased mental health and behavioral problems when compared to children who never saw their parents argue.

Children learn compassion and how to use love and kindness to solve disagreements. They also learn that disagreeing with somebody you love is okay. It doesn't mean they'll stop loving you.

Not exactly the same as showing a united front or whatever but relevant in my opinion.

The last thing I'll add is that it's also pretty well established that children respond best to clear boundaries. They need to clearly understand what behavior is acceptable and unacceptable, i.e., what is safe and unsafe. If dad gets angry about something and mom doesn't care, it's confusing for them and leads, again, to emotional instability.

So yeah, it's complicated. It can be good to disagree and model healthy resolution behavior, but at the end of the day, both (all) authority figures do need to set similar boundaries.