'anti-monarchically' is formed from the prefix 'anti-' (from Greek 'anti' meaning 'against') combined with 'monarchical' (from 'monarch', ultimately from Greek 'monarkhēs', where 'monos' meant 'single' and 'arkhein' meant 'to rule'), plus the adverbial suffix '-ly'.

'monarch' entered English via Latin and Old French from Greek 'monarkhēs'; 'monarchical' developed in Late Middle English/Early Modern English; the adverb 'monarchically' was formed with '-ly', and 'anti-' was prefixed in modern usage to form 'anti-monarchically'.

Originally 'monarch' referred to a 'single ruler'; over time it came to denote the institution or office of a sovereign (hereditary ruler), and with the prefix 'anti-' the compound now denotes opposition to that institution or principle.