In the direction or manner of an ant, or proceeding in the way that ants move.
Combination of 'ant' (the insect) and the adverbial suffix '-wise' (meaning 'in the manner of'), following the pattern of words like 'clockwise' and 'lengthwise'.
The suffix '-wise' is totally productive in English—you can stick it on almost anything to mean 'in that direction or way'—so antwise humorously describes moving with the organized, methodical pace ants are famous for.
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