'anti-domestically' is formed from the prefix 'anti-' (from Greek 'anti-' meaning 'against') + 'domestic' (from Latin 'domesticus' derived from 'domus' meaning 'house') with the adverbial suffix '-ally'.

The element 'domestic' comes from Latin 'domesticus' ('of the house'), passed into Middle English via Old French; the productive prefix 'anti-' (Greek) was later attached to form 'anti-domestic' in modern English, and the adjective took the adverbial suffix to become 'anti-domestically'.

Initially, 'domestic' meant 'of the house' or 'relating to the home'; with the addition of 'anti-' the compound took on the sense 'opposed to home/household/domestic arrangements', and the adverb now means 'in a manner opposing domesticity or domestic policies'.