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If the two positions are incompatible, then a person who accepts one should reject the other. So:
My favorite is the sneering pro-choice activist who claims that the pro-life advocate should also care about children's safety and defend their rights after they are born. A cause they advocate. Yet in the face of their own assertion that opposition to abortion rights is consistent with liberal child welfare policies, they support those policies while also supporting abortion rights.
The important point is that they do this mainly to confuse people, and to drive their adversaries onto the wrong side of low stakes issues out of spite.