The Australian-English phrase to drink with the flies means to drink by oneself in a public house, which is regarded as an unsociable attitude. The image is that the solitary drinker has no other companions than the flies.
—Cf. also the Australian-English phrase Jimmy Woodser, which designates:
– a person drinking alone at a bar;
– a drink taken alone.

In The Australian Language (Sydney and London: Angus and Robertson Ltd., 1945), about the larrikin’s immense “influence on Australian speech as a whole”, Sidney John Baker (1912-1976) explained the cultural significance of the phrases to drink with the flies and Jimmy Woodser:

Several of his [i.e., the larrikin’s] characteristics should be underlined. Primarily there is his toughness, recklessness and brutality. Then there is his instinctive desire to be one of a company. Much of our large drinking vocabulary built round the shout and the companionship it implies, together with the Australian’s contempt for the Jimmy Woodser or the man who drinks with the flies, is closely linked with the larrikin’s masculine idea of mateship.

Both the facts that drinking alone is considered unsocial and that the phrase to drink with the flies refers to the flies as sole companions are illustrated by the following story, published in Smith’s Weekly (Sydney, New South Wales) of Saturday 6^th July 1935—Adelaide is the capital and chief port of the state of South Australia:

WHERE MEN DON’T DRINK ALONE
Tale of a man who did!

This story is told in Adelaide by men who come South from Central Australia. It is about a prospector who struck it rich three years ago, and called for a drink at an Alice Springs hotel.
Noticing that he drank alone a wag asked, “Aren’t you going to shout?” Among those present were a few officials and the local J.P. The prospector lifted his glass and replied, “I made it myself and I am going to spend it on myself. I am drinking with the flies.”
Then he had two more drinks, also with those winged pests of the north.
Observing the looks of disapproval he wandered to the door, drew a revolver, and fired into the tyres of a car parked outside. The car settled on its rims. “I made it myself and I am going to spend it on myself,” he said as he wandered back. He probably estimated that he would have to pay for the tyres.
Two or three more drinks alone, and each drink was accompanied with the refrain, “I made it myself and I am going to spend it on myself.”
By this time the Alice Springs beer was doing the work of all beer. The prospector went to the door again. A small galvanised water tank was his target this time. A few shots, and the tank was emptying itself through a number of holes.
Again he returned to the bar muttering, “I made it myself and I am going to spend it on myself.”
He was eligible for arrest when he journeyed to the door for the third time. He noticed some small cuttings that had been planted round the pub to beautify the surroundings. They seemed to annoy him—these splashes of green in the dry street.
One by one he tore them up and threw them over his shoulder, accompanying each action with, “I made it myself and I am going to spend it on myself.”
And so he celebrated, while the habitues of the hotel watched his progress.
It was towards his last approach to the door that the local policeman took charge of him, and next morning he appeared to answer for his misdeeds before the local court.
The J.P. was in a bright mood to begin with.
“Why did you fire at the motor car?” he asked.
“I don’t know, sir.”
“Well, we will forgive you for that,” said the J.P. And the offender was also forgiven for shooting at the tank and pulling up the trees.
Then there was a rustle in the court as the J.P. said, “Why did you drink alone?”
“Well, I made it myself and I was spending it on myself,” replied the prospector.
The J.P. looked very grave. “£40 or two months,” he said. This merely for drinking alone. But it certainly covered the other damage.
“It is one of the things we never forgive in the North,” concluded the J.P.—“this drinking alone.”

These are the earliest occurrences of the phrase that I have found, in chronological order:

1-: From the column Snap Shots, by ‘Gatling’, published in the Tasmanian Democrat (Launceston, Tasmania) of Friday 25^th March 1898:

There is a good story going the rounds about a sovereign. A West Coaster dropped into one of our city pubs, and being one of those who hate “drinking with the flies,” asked a dead-beat, who was scanning the morning papers, to join him. A sovereign was tendered, but change being short the West Coaster was asked to pay when he called again. When the landlord’s back was turned, the dead-beat borrowed the sovereign, and with a lordly air commanded the landlord to “fill ’em again.” The ubiquitous sov. was tendered. The landlord this time transferred it to his pocket, and handed over 12s 6d to the dead-beat, remarking “That will clean the slate, old fellow. You have owed 7s for the last six months.” There was a sultry ten minutes between the landlord, stranger, and dead-beat, but boniface ¹ scored.

¹ The name Boniface designates the keeper of an inn, hotel or public house. It originated in Boniface, the name of the jovial innkeeper in The Beaux Stratagem (1707), a comedy by the Irish playwright George Farquhar (1677 or 1678 – 1707).

2-: From The Bulletin (Sydney, New South Wales) of Saturday 15^th April 1899:

“WITH THE FLIES.”

A bushman thinks the meanest man—on earth, at all events—
Is he who has a liquor “with the flies;”
They’ll forgive a man who robs them, but not a man consents
To look at him who boozes “with the flies.”
He may not know a single soul of those who fill the bar;
He may be but a stranger who has travelled from afar;
But let him liquor solus, and the lads say, “There you are!
He’s a crook ’un, for he’s drinkin’ with the flies.”
Stan. Warrington.

3-: From The Inverell Argus (Inverell, New South Wales) of Tuesday 30^th May 1899:

FEDERATION.
THE VIEWS OF COW FLAT.

A meeting of the citizens of Cow Flat was held on Thursday last in the back room of Mr. Michael Finnegan’s “Snake Juice Inn,” for the purpose of discussing the all absorbing question of Federation, and finally deciding which direction the collective vote of this important centre should be given at the approaching referendum. […]
[…]
Mr. Finnegan said he was well able to talk on this great question as he was a deep reader, having read every Deadwood Dick ² in print. In his mind the greatest point of interest about the Bill to the Cow Flat people was the federal city question. We should put in our claim at once, for although outside the prescribed limit of 100 miles from the sea coast we had a claim on another score, viz., that we were 100 miles exactly from civilization. Should Cow Flat become the federal city, he would shout.
Mr. MacSquirt: Garn ³, you never shouted in your life; you drink with the flies.
[…]
Mr. Finnegan: I was going on to say I would shout three cheers for Queen Victoria and Brian Boru ⁴. Should Melbourne become the federal city, you here in New South Wales can close up.

² Deadwood Dick is the name of a character who appears in a series of dime novels by the U.S. author Edward Lytton Wheeler (1854 or 1855 – 1885).
³ The interjection garn, representing a regional pronunciation of go on, is used to express disbelief or ridicule of a statement.
⁴ Brian Boru (circa 941 – 1014) was the last High King of Ireland.

4-: From The Mount Lyell Standard and Strahan Gazette (Queenstown, Tasmania) of Wednesday 14^th March 1900:

A VISIT TO ZEEHAN ⁵.
FLEETING IMPRESSIONS.
(“Standard” Special.)

[…]
Zeehan hotels do not differ from other West Coast hotels, for their beer is also sixpence. Their customers, however, have many peculiarities. It is by no means a rare sight to see an individual drinking with the flies in a bar parlor, and it is noticeable that the individual seems to enjoy the flies’ company. At any rate he betrays no inclination to drink with anybody else but the flies.

⁵ Zeehan is a town on the west coast of Tasmania.