Green’s Dictionary of Slang

(Aus.) in the game of two-up, one who favours betting on ‘tails’.

1924		(con. WWI) A.G. Pretty Gloss. Sl. [...] in the A.I.F. 1921–1924 (rev. t/s) n.p.: tailie. A man who backs tails in a game of ‘two up.’.
1949		L. Glassop Lucky Palmer 159: Might be a tail trot to-night and I’m a tailie.
1977		(con. 1941) R. Beilby Gunner 296: He was having trouble finding a taker for on a tailies day like this most men were reluctant to spin.