'anti-communistic' is formed from the prefix 'anti-' (from Greek 'anti' meaning 'against') combined with 'communistic', the adjectival form of 'communism' (from French 'communisme' and Latin 'communis' meaning 'common').

'communis' (Latin, 'common') → French 'commun'/'communisme' → English 'communism' (19th century) → adjective 'communistic' → compound with prefix 'anti-' in English to form 'anti-communistic' (20th century usage in political contexts).

Initially the elements meant 'against' + 'common/shared' (via 'communis'), and the compound has come to mean specifically 'against the political ideology of communism'.