The banana-growing nations of Central America were called “banana republics’ in the 1900s. Short story writer O. Henry (1862-1910) used the term “banana republic” in several of his stories and is given credit for coining and popularizing the term.
 
“Banana republic” quickly gained a derogatory meaning of a small country run by a dictator. “Banana republic” now can apply to any country (not only a banana-growing country in Central America); the term can also apply to states, cities and corporations that are corrupt and are poorly run.
 
“The only difference between the U.S. and a banana republic is we don’t grow bananas” is a related saying. 
     
     
Wikipedia: Banana republic
Banana Republic is a pejorative term for a country that is politically unstable, dependent on limited agriculture (e.g. bananas), and ruled by a small, self-elected, wealthy, and corrupt clique. It is most commonly used for countries in Central America and Africa such as El Salvador, Belize, Grenada, Nicaragua, Honduras, Guatemala, Ethiopia and South Africa. In some cases, these nations have kept the government structures that were modeled after the colonial Spanish ruling clique, with a small, largely leisure class on the top, and a large, poorly educated and poorly paid working class of peons, though it might have the (fake) trappings of modernity (such as styling itself a republic with a president etc.)
 
Frequently the subject of mockery and humour, and usually presided over by a dictatorial military junta that exaggerates its own power and importance—“the epaulettes of a banana republic generalissimo” are proverbially of considerable size, usually portrayed in satire with a pair of mops—a banana republic also typically has large wealth inequities, poor infrastructure, poor schools, a “backward” economy, low capital spending, a reliance on foreign capital and money printing, budget deficits, and a weakening currency. Banana republics are typically also highly prone to revolutions and coups.
   
Background
The term was originally invented as a very direct reference to a “servile dictatorship” which abetted (or directly supported in return for kickbacks) the exploitation of large-scale plantation agriculture (usually banana). The term was coined by the American author O. Henry in his 1904 book of linked short stories, “Cabbages and Kings”, set in the fictional “Anchuria”, which was based on his 1896-97 stay in Honduras.
 
It was in Honduras that the United Fruit, the Standard Fruit, and Sam Zemurray’s Cuyamel Fruit companies dominated the country’s key banana export sector and support sectors such as railways. The United Fruit Company was nicknamed “The Octopus” (El Pulpo) for its willingness to involve itself in politics, sometimes violently. In 1910, Zemurray hired a gang of armed thugs including Lee Christmas from New Orleans to stage a coup in Honduras to obtain beneficial treatment from the new government. Zemurray would 22 years later take over United Fruit in a hostile bid.
 
Four decades later, the directors of United Fruit played a role in convincing the Truman and Eisenhower administrations that the government of Colonel Arbenz in Guatemala was secretly pro-Soviet, thus contributing to the CIA’s decision to assist in overthrowing Arbenz’s government in 1954 (see Operation PBSUCCESS). Pablo Neruda would later denounce the dominance of foreign-owned banana producers in the politics of several Latin American countries in a poem titled “La United Fruit Co”.
 
Modern usage
In modern usage the term has come to be used to describe a generally unstable or “backward” dictatorial regime, especially one where elections are often fraudulent and corruption is rife. By extension, the word is occasionally applied to governments where a strong leader hands out appointments and advantages to friends and supporters, without much consideration for the law. A banana republic can also be used to describe a country where a large part of its economy and politics are controlled by foreign powers or even corporations, e.g. Myanmar.
 
Some Central American countries, like Belize, that export bananas to a specific client or set of clients as part of a continual agreement or previously agreed price are not banana republics in the way the phrase is defined above.
 
Wikipedia: Banana Republic (clothing retailer)
Banana Republic (Republique Bananique in French) is a chain of mainly United States based clothing stores founded by Mel and Patricia Ziegler in 1978 as a travel-themed clothing company. It was bought by Gap in 1983. The company has over 500 stores.
 
Google Books
The Yale Book of Quotations
Edited by Fred R. Shapiro
New Haven, CT: Yale University Press
2006
Pg. 355:
O. Henry (William Sydney Porter)
U.S. short story writer, 1862-1910
“In the consultation of this small, maritime banana republic was a forgotten section that provided for the maintenance of a navy.”
Cabbages and Kings ch. 8 (1904). Appears to be the coinage of banana republic, previously thought to trace back to 1935.
   
Wikipedia: O. Henry
O. Henry was the pen name of American writer William Sydney Porter (September 11, 1862 – June 5, 1910). O. Henry’s short stories are well known for their wit, wordplay, warm characterization and clever twist endings.
   
Merriam-Webster Dictionary
Main Entry: banana republic
Function: noun
Date: 1904
: a small dependent country usually of the tropics ; especially : one run despotically
 
(Oxford English Dictionary)
banana republic n. colloq. applied to a small state, esp. in central America, whose economy is almost entirely dependent on its fruit-exporting trade.
1935 Esquire July 70/1 We strung along with Major Brown on the inhuman aspects of war in the *banana republics.
1949 A. KOESTLER Promise & Fulfilm. xiii. 143 The somewhat jerky behaviour displayed by the Central American banana republics.
   
4 January 1902, Seattle (WA) Daily Times, pg. 16, col. 6:
‘The Flag Paramount,” by O. Henry, is a picturesque story of a Banana republic, founded on a very original idea.
(Ainslee’s Magazine for January—ed.)
 
Google Books
What I Saw in the Tropics
By Henry Clemens Pearson
New York, NY: India Rubber Publishing Company
1906
Pg. 185:
THE FIRST SIGHT OF COSTA RICA, THE BANANA REPUBLIC
(...)
Not only is Costa Rica called the Banana Republic, but Port Limon is a banana town, and we fully appreciated it when we saw the train loads of green fruit run out upon the piers, the huge bunches dumped upon rubber conveying belts and carried smoothly into the holds of the waiting steamships.
 
Chronicling America
10 March 1906, Pensacola (FL) Journal, pg. 4, col. 2:
We refer to the countries to the south of us as “banana republics,” as “volcanic nations,” and we are sadly underestimating what these countries represent.
(“Extent of Spanish Speaking World” in Memphis Commercial-Appeal—ed.)
 
Chronicling America
6 May 1906, Washington (DC) Times, “The Flag Paramount” by O. Henry,  magazine section, pg. 2, col. 1:
In the constitution of this small, maritime banana republic was a forgotten section providing for the maintenance of a navy.
Pg. 11, col. 5:
Copyrighted, 1902, by Ainslee Magazine Co.
 
Chronicling America
13 May 1906, Washington (DC) Times, “Money Maze” by O. Henry, magazine section, pg. 3, col. 6:
At that time we had a treaty with about every foreign country except Belgium, and that banana republic, Anchuria.
Pg. 4, col. 7:
Copyright, 1902, by the Ainslee Magazine Company.
 
Chronicling America
20 May 1906, Washington (DC) Times, “Rouge et Noir” by O. Henry,  magazine section, pg. 3, col. 6:
The banana republic of Costaragua has, practically, two capitals.
Pg. 11, col. 7:
Copyrighted, 1901, by Ainslee Magazine Co.
     
OCLC WorldCat record
Gypsying through Central America : [a biographical narrative, or diary, of Eugene Cunningham and Norman Hartman’s journey through the “banana republic” of Central America]
Author: Eugene Cunningham
Publisher: London, [Eng.] : T.F. Unwin, 1922.
Edition/Format: Book : Biography : English
     
OCLC WorldCat record
Honduras the banana republic
Author: John W Green; Herbert E. Budek Films and Slides (Firm)
Publisher: Herbert E. Budek Co., 1963.
Edition/Format: Visual material : Filmstrip : EnglishView all editions and formats
Summary: A survey of Honduras. Describes the geographical features and the historical background; notes important features of various cities and ports; and examines the way of life of the people, transportation within the country, and the chief products.
 
OCLC WorldCat record
Honduras : the making of a banana republic
Author: Alison Acker
Publisher: Toronto : Between the Lines, 1988.
Edition/Format: Book : English