'antihygienically' is formed in English from the prefix 'anti-' (from Greek 'anti' meaning 'against') + 'hygienic' (from Greek 'hygieinós'/'hygieia' relating to health) + the adverbial suffix '-ally'.

'anti-' entered English via Latin/Old French from Greek 'anti' meaning 'against'. 'Hygienic' comes via French 'hygiénique' and modern scientific/medical Latin from Greek 'hygieinós' (from 'Hygieia', the personification/goddess of health). These elements combined in modern English to form 'antihygienic' and then 'antihygienically'.

Initially the elements meant 'against' + 'healthful/relating to health'; together they evolved to mean 'against hygiene' or 'unsanitary', and the adverbial form now means 'in an unsanitary manner'.