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In Australian-English slang, the noun big spit designates vomiting, and the phrase to go for the big spit (and variants) means: to vomit.

The following, for example, is from Behind the titans, two football brains work overtime, by Roy Masters, published in The Sydney Morning Herald (Sydney, New South Wales, Australia) of Monday 23^rd May 1994 [page 41, column 4]—the Australian Rugby-League player Allan Langer (born 1966) was then the captain of the Brisbane Broncos, a club based in Red Hill, Brisbane, Queensland:

A few minutes before game-time is the only occasion anyone knows how he [i.e., Allan Langer] is feeling. If it’s a big game, he vomits until there is nothing left inside him.
[…]
Broncos boss John Ribot, linking Langer’s past vomiting to the club’s past victories, said “his chief executive gets a very comforting feeling when Alf has the big spit.”

—Cf. also the phrases to shoot the cat and to drive the porcelain bus.

These are, in chronological order, the earliest occurrences of the noun big spit and of the phrase to go for the big spit that I have found:

1-: From A Ripley Story, published in The Warwick Daily News (Warwick, Queensland, Australia) of Saturday 20^th March 1943 [page 5, column 7]:

This story, unusual but true, had its origin at Sussex Inlet last week. Three men, visitors to the Inlet, hired a launch from Mr. A. Glanville to spend a day on outside fishing (says the Shoalhaven News, Nowra).
On their return that evening a nasty sea had worked up on the bar, and the launch was overturned. Fortunately for one of the fishermen, who was unable to swim, the tide was coming in, and with the assistance of his two mates, he was able to make the shore.
Later events were to prove that the non-swimmer was a careful sort of cove. Anticipating that he might experience an attack of the “big spit,” or in other works [sic] contract sea sickness, he carefully “de-teethed” and rolled them in a handkerchief, together with his money, and placed them for safety’s sake in his trousers pocket.

2-: From the column Club Capers, by ‘Mortar Bomb’, published in The Biz (Sydney, New South Wales, Australia) of Thursday 27^th March 1952 [page 8, column 5]:

Joeosci really went to town on Saturday, when for the first time he went for the “big spit.” Following up with a great four-round exhibition on the football field. Joesci [sic] really got on the wrong end of a straight left, and we reckoned he was about to burst into a samba, the way his leg wobbled.

3-: From A World of Our Own (London: James Barrie, 1955), by the Australian author Gerald Marcus Glaskin (1923-2000) [page 222]:

She could hear him being violently ill somewhere behind the stone grotto. After a while they came back and there was a stain down his shirt and trousers. Rose was cleaning him up  with a handkerchief. He was staring at her with dull stupid eyes that refused to focus, and clinging on to her shoulders with his hands. He didn’t seem to recognize her and she was telling him he was a beaut, goin’ for the big spit so early in the night!

4-: From the following, by ‘The Old Gentleman’, published in the Royal Australian Navy News (Sydney, New South Wales, Australia) of Friday 20^th March 1959 [page 4, columns 1, 2 & 3]:

The Big Spit
ACROSS BASS STRAIT IN WAGGA

Down in the Strait
The waves rolled high,
The waves rolled wide
And the “Big Spit” starters lined the side.
They threw it high and
They threw it low.
They tossed it overside in the sunset glow.
The cooks [sic] looked out and chuckled with glee,
And said, “If this keeps up it will do me.
They’ll need no meals, these scurvy knaves—
They’ll only heave it back to the angry waves.”
So he gloated with sadistic pride.
Then gulped and rushed to join
With the other “Big Spits” lining “Wagga’s” lee side.
[&c.]

5-: From You Only Live Twice (New York: Signet Books, published by The New American Library of World Literature, Inc., 1965), by the British author Ian Fleming (1908-1964) [chapter 5: Magic 44, page 37]—the speaker is Dikko Henderson, an Australian agent in Japan:

“I’ve got myself a proper futsuka-yoi—honourable hangover. Mouth like a vulture’s crutch. Soon as we got home from that lousy cat house, I had to go for the big spit.”