It is foolish to view realism and idealism as incompatible or to consider our power and wealth as encumbered by the demands of justice, morality, and conscience.
I was taught that the search for truth and the search for justice are not incompatible and are, in fact, essential.
Being in love with a lot of people is incompatible with a stable family life.
Disagreements between incompatible beliefs cannot be settled by reasoned argument because reasoned argument is drummed out of those trained in religion from the cradle.