How can one maintain ties with half-siblings whom one does not like, some of whom live nearby and some far away, with the bare minimum? And is it sufficient to call them once a month?

Maintaining ties with brothers and sisters is obligatory, and it is not permissible to sever them. Problems or lack of affection do not justify abandoning this duty. Maintain ties with them in obedience to Allah's command, to spite Satan, and to avoid the escalation of problems. Maintaining kinship ties is not a matter of reciprocity, and the customary practice among people is the reference for determining what it entails.