oil and water; incompatible by nature
Proverb meaning two things or people are fundamentally incompatible, like ice and hot charcoal. Often used to describe irreconcilable differences in personality, opinion, or nature.
See also: 相容れない (あいいれない)
He and I are like oil and water; it's impossible for us to work together.
The two opinions were completely incompatible and could not be reconciled.
氷炭相容れず is a fixed proverb emphasizing fundamental incompatibility, while 相容れない is a more general adjective meaning 'incompatible' or 'cannot coexist'.
From Chinese classical literature, literally 'ice and charcoal cannot coexist', symbolizing two things of opposite nature that cannot be together.