IPA: /ˌɪrɪˈkɒnsaɪləbli/
KK: /ɪˌrɛkənˈsaɪləbli/
In a way that shows two things cannot be made compatible or brought to agreement.
The two sides argued irreconcilably, refusing to find any common ground.
Irreconcilably → It is formed from "in-" (meaning not), "reconciliare" (meaning to bring together or restore to friendship), and "-ably" (meaning in a manner). The word describes a state of being unable to be brought together or restored to agreement.
Think of 'not' ('in-') being able to 'bring together' ('reconciliare') in any way — that's why irreconcilably means unable to reach an agreement.