Take any well-divided political issue, and you'll find people on both sides building up detailed stories that show what it is exactly that ought to convince any reasonable person, but fails to convince their opponents due to their ideological bias. Such stories are almost always wrong. Typically they do one or several of:
(i) exaggerate the evidence or misrepresent its degree of uncertainty;
(ii) ignore conflicting evidence to the other direction;
(iii) tacitly assume a host of underlying convictions that are only obvious to your side;
(iv) ignore any number of ways the other side could find to explain your evidence without changing their beliefs, not all of them contrived.

A good analysis of what it is that makes politics (or at least American politics) a mind killer. In fact, worse than a mind killer. The habit of convincing yourself that those who disagree with you are subrational (and intellectually dishonest to boot) is the community killer - it is the first step in a rationalization of disenfranchisement.

Are there other subjects besides politics which lead to the same dehumanization of the people who disagree? I think so. One sees it frequently in theological disputes, pretty often in ethical disputes, and occasionally when discussing interactions between the sexes. But very rarely in discussions of the arts, music, spectator sports teams, grammar, and even nutritional practices - even though tribalism is common enough in these areas, no one tries to paint their opponents as either fools or knaves. Why the difference - is it just because these topics are less important than politics?

According to Aumann, we should be able to agree to disagree only if one of the following is the case:

• We have different priors (or different fundamental values)
• One of us is irrational
• We don't trust each other to report facts and beliefs truthfully
• We just don't talk enough.

So, if Aumann is to be believed, in those cases where we do talk enough, and in which we claim to share priors and fundamental values, disagreement is likely to turn nasty.

ETA: HT to Plasmon for pointing out the counter-intuitive fact that disagreement may be less nasty when divergence of fundamental values is acknowledged.