The Tigers went to stumps on day three at 1-41 chasing 429 for victory, with Jake Weatherald (26no) and Tim Ward (9no) at the crease.
Tasmania lost Nivethan Radhakrishnan for four early in their run chase when the opener was bowled by Nathan McAndrew.
The Tigers have a big mountain to climb to win, or even draw, the match.
South Australia took control of the contest by posting 6(dec)-398 and skittling Tasmania for 203.
Things were looking even more dire for Tasmania as the Redbacks moved to 0-117 and 3-211 in their second innings.
But the Tigers' attack hit back hard, sparking a collapse of 6-22 before South Australia declared at 9-233.
Who needs two hands anyway right? — Tasmanian Tigers (@TasmanianTigers) pic.twitter.com/bQJlnBqjojDecember 8, 2024
Conor McInerney (75 off 73 balls), Henry Hunt (31 off 57) and Jason Sangha (61 off 92) had set an excellent platform for South Australia before Tasmania clicked into gear.
Spinner Matt Kuhnemann, who has his eyes on a Test recall for the tour of Sri Lanka, was the chief destroyer with 4-67.
Kuhnemann produced a spectacular one-handed catch off his own bowling to remove Hunt, and bowled Liam Scott and Jake Fraser-McGurk as they attempted to heave him for six.
Nathan McAndrew became his fourth scalp, with paceman Gabe Bell (2-41) the only other multiple wicket-taker for the Tigers.
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