John Hotten, A Dictionary of Modern Slang, Cant, and Vulgar Words (1859) has this entry for "shirty":
SHIRTY, ill tempered or cross. When one person makes another in an ill humour he is said to have "got his SHIRT out."
It is not entirely clear from this description whether the the word "he" refers to the person who puts the other person in an ill humor or to the person whose humor is soured.
Although the expression "got [one's] shirt out" seems to have originated in the British Isles, it reached Australia by the late 1850s. The National Library of Australia Trove of old newspapers reports 15 matches for "got his shirt out" in the period from 1859 to q937. Here are the three earliest matches.
From "Public Meeting at the Princess's Theatre," in the [Melbourne, Victoria] Argus (May 25, 1859):
Mr. Manuel made an attempt to address the meeting again. He said he thought they were gentlemen, but found he was mistaken. (A voice—"You're a loafer," and tremendous uproar.)
(At this period of the meeting, Mr. Osborne passed in front of the Chairman, an imperfection in his nether garments eliciting the remark, " Hear him ; he's got his shirt out already.")
Mr. Wilson Gray then came forward and read his amendment a second time, amid loud cheering and a few hisses.
This instance suggests that "to have [one's] shirt out" was already a popular slang phrase or idiom, and that the anonymous speaker was making a jest based on the associated meaning.
From an advertisement for R.T. Parnell's Grand Betting Company in the Alexandra [Victoria] Times (August 13, 1868):
Ginger Chops is a likely looking light chesnut stallion, of a light build, and has had too much hard riding lately, being much in requisition as a hack. While training for this race he has got completely bogged on the 3rd, 4th, 19th, 45th, 48th, 60th, 75th, 76th, and 184th clause swamps, one after the other, where he has been floundering for the last three weeks until he has strained his hind quarters, and got his shirt out. This misfortune arose from his silly attempt to show his points and head "Bill of Costs" and an old Government hack called "Higginbotham." But for this the betting might have been fair on "Ginger."
This item is part of an extended assessment of a series of politicians disguised as racehorses. The factual counterparts of the metaphorical descriptions are lost on me.
From "News and Notes," in the [Brisbane, Queensland] Queenslander (November 13, 1869):
It has become a regular personal quarrel—a war of invective and bitter abuse. I must say that the Government side, considering who and what some who sit on the same side as Ministers really are, and how prone they might be supposed to be to a "ruction" conduct themselves with great forbearance and moderation. A few evenings since though, the member for the Northern Gold Fields, Mr. Hoskins, in replying to Mr. Parkes, "got his shirt out," to use a vulgar expression—which by the way, for the information of the innocents, I may explain means no more harm than this, that a person gets into such a passion that he incontinently pulls off his shirt to fight. The expression, however, is generally used in a figurative sense, and so it is now. Hoskins, adverting to the stern denouncements of Parkes against all corruption, &c., reminded him that he, when Colonial Secretary, on £2000 a year, got a lady member of his own family to earn £20 from the public treasury for translating one or two French pamphlets—which plenty of people not connected with the Minister would have been glad to have done for one-half that, and which indeed, ought to have been done by one of the public servants. Well, the affair
was paltry enough, viewed any way, but the attack was justified by the undoubted fact that Mr. Parkes would have been the first to cry out in a similar case. However, he got into a great rage, and made a most indignant speech, declaring that he would never forget this treatment as long as he lived.
The author of this note (identified as "a Sydney man") conveniently explains that the expression arose in connection with removing one's shirt prior to fighting but subsequently applied in a metaphorical sense to heated disputation or ill feeling.
As "a Sydney man" writing in 1869 informs us, the expression "got his shirt out" referred to removing one's shirt prior to a physical fight—presumably in order to avoid having it bloodied or torn in the course of the ensuing fisticuffs. Pictures of prizefights from this period (and earlier) frequently show the combatants stripped to the waist as they battle in the ring.
The expression was adapted to idiomatic use as a way to indicate that someone had taken umbrage at something someone else had said and was spoiling for a (metaphorical) fight. It follows that "getting [one's] shirt out" is something that the shirt wearer did, upon provocation by someone else. This contrasts with the (admittedly ambiguous) comment that Hotten provides in his 1859 slang dictionary.
The expression probably originated in Britain, from which nineteenth-century colonists then transported it to Australia. It seems never to have caught on in the United States, as none of the instances of "got his shirt out" in the relevant sense that an Elephind search turns up are from that country.