If The Alliance is the Five-Man Band of political entities, The Republic will likely be The Hero (or just thinks it is), The Lancer, or The Big Guy. The Republic is the close relative of The Federation. If it is not part of The Alliance, it is usually friendly to it. It is almost always enemies with The Empire. Like The Alliance, it sometimes forms out of La Résistance. This government, like The Good Kingdom, is almost never evil. If it is, it's nearly always because of some military coup or a group of corrupt bureaucrats. Sometimes, The Republic ends up becoming The Empire because of this.
The Republic is a far more tight-knit nation than The Federation, usually acting like one nation rather than separate countries, but shares most of its characteristics. If The Republic and The Federation are both in the same Verse, they may not always be on the friendliest of terms. It is often based in most ways on the modern United States, and generally some kind of humanitarian, democratic capitalist society.
In terms of technology, The Republic is often the most highly advanced group in The 'Verse, possessing the most intensive research programs and best-equipped military (aside from The Empire). If it is pitted against The Empire, they will usually engage in a Lensman Arms Race, and will often be militarily similar. The Republic is sometimes more prone to open warfare with The Empire. If there is no Empire, The Federation and The Republic may act as The Empire for each other.
Keep in mind that republics in Real Life can differ much. A republic is not synonymous to a democracy. There are democratic republics (what most people think of first), but Venice, Genoa, and the Netherlands were oligarchic republics, and many dictatorships officially are republics too, even those where the dictator manages to make his offspring his successor. The only common denominator for republics is that they do not have a monarch or other formally titled aristocracy as their ruler. By default, non-democratic republics tend to be oligarchies—either stratocracy (military rule), aristocracies or plutocracies, or "spookokracies" led by the Secret Police. At worst, a Republic will be an Intellectually Supported Tyranny—sporting a violent, totalitarian ideology.
The Trope Namer can be attributed as The Galactic Republic, usually referred to as just The Republic. However, in trope terms it's more like The Federation. Not to be confused with the People's Republic of Tyranny, or Plato's work of political philosophy. See also Good Republic, Evil Empire when The Republic is cast in opposition to The Empire.
- Alderamin on the Sky: The Republic of Kioka, which is notably depicted from the point of view of The Empire. Kioka is technologically advanced and advancing further due in large part to its extensive civil liberties, whereas the protagonists' Katvarna Empire is stagnating due to political corruption and religious dogmatism. Main character Ikta Solork's mentor ended up fleeing to the Republic in the backstory because his research ran into opposition from the Empire's upper classes.
- Star Wars:
- The lore starts with the Galactic Republic, which stood for thousands of years before being corrupted from the inside and transformed into the Galactic Empire.
- The Force Awakens shows that in the wake of the Imperial defeat a few years after the Battle of Endor, a "New Republic" was set up to take its place. However, a remnant of the Empire exists in the form of the First Order, who continue to fight the successor to the original Rebel Alliance, The Resistance. About halfway into the movie, the New Republic's current capital (and the entire fleet surrounding it) gets blown up by the First Order.
- A Song of Ice and Fire has the Nine Free Cities, as well as the slave republics of Yunkai, Mereen, and Astapor. Generally speaking, these states give their citizens more freedoms than the Seven Kingdoms of Westeros do, but at the price of legalized slavery (except in Braavos, founded by slaves who revolted, freed themselves and formed a country, with slavery naturally banned).
- Honor Harrington: A wide variety of examples have been seen so far:
- The People's Republic of Haven used to be one of these in the Back Story, and side stories taking place a few centuries back such as the Manticore Ascendant depict Haven as being one of the most powerful examples in the galaxy. Later books in the main series have Haven reverting back to being a Republic, and having to deal with the various problems of the transition and lingering hostility with the Star Kingdom of Manticore.
- The Solarian League similarly used to be one of these before it gradually changed into something else due to centuries of complacency and increasing power of the Bureaucracy. A number of the League's older member worlds are still indicated to be examples unto themselves, however.
- The Star Kingdom of Manticore (and later The Star Empire of Manticore, despite the name, tends to be a closer example than most of the powers claiming the title of "Republic", being a Constitutional Monarchy modeled after the United Kingdom of of Great Britain and Northern Ireland. The Crown is hereditary, and has some very specific powers, but otherwise everything is run by the House of Lords (who inherit their seats) and the House of Commons (who are not allowed to hold hereditary title and are voted for by the populace).
- Military science fiction Victoria has the Northern Confederation, a reactionary version with weak central government and strong elements of direct democracy—in some respects a Jeffersonian homesteader republic, but updated for the new era century with cold fusion and a ban on color television. Most of their enemies are various flavors of the People's Republic of Tyranny.
- In Shadow of the Conqueror, Hamahra (the former seat of the Dawn Empire) has moved away from the monarchies of its past and become a republic ruled by a democratically elected Senate.
- In Lone Huntress, the vast majority of humankind live on planets (or space habitats, or ships) under the authority of the Federation, born after a secret Alien Invasion resulted in a bloodsoaked mass revolution and eventual formation of a government based on the theory that, since the Ancient Evil relies on the fostering of tyrannical regimes, freedom is ensured by protecting liberty, rather than choosing between them. The bad guys tend to either be pirates, corrupt business types and government officials, and of course the Fey.
- Game of Thrones adapts the republics of the Free Cities and Slaver's Bay more or less faithfully.
- In Star Trek: Deep Space Nine, Bajor is established to be a republic with an elected First Minister. However, it has elements of The Theocracy because of how much influence the Kai (essentially the Bajoran pope) has over the people; this is best demonstrated when Kai Winn is appointed as an interim replacement for a recently-deceased First Minister and tries to get herself permanently elected to the position until La Résistance hero Shakaar throws in his hat and gets elected instead.
- Pathfinder has a number of examples, but the most archetypal and iconic is the Andoran Republic. Founded by Taldoran and Chelish rebels who were tired of living under tyranny. The state they've built is a fairly clear stand-in for the USA, ruled by a People's Council (which includes congressmen and representatives of the various trade guilds), and a Supreme Elect (president). They're one of the most predominantly "good" nations in the setting.
- Crusader Kings II features merchant republics (there are also ordinary republics, but those are not playable), including Venice, Genoa, Pisa, and any others you might care to invent (just declare a mayor of a coastal county to be the lord of a ducal sized domain). They provide fewer soldiers, but produce more tax revenue than regular ducal domains. There's also a Game of Thrones mod that allows you to play as the various republics of Essos.
- Democratic empires in Stellaris, two of the basic types (Theocratic Republic and Military Republic) have it in their name. Oligarchies may also count as they have elections, it's just they're once every forty years and only the elite are allowed to vote.
- Ace Combat:
- Ace Combat 6: Fires of Liberation has the Republic of Emmeria. Unlike most playable factions in the series, Emmeria is a single unified nation, not a multinational alliance or a federation. In game lore shows that Emmeria was once a kingdom, but relinquished its monarchy sometime in the past. The former royalty holds no official power and the national castle is used as a history museum.
- In Ace Combat Zero: The Belkan War, you play as a mercenary pilot in the Republic of Ustio Air Force.
- In Mahu's "Second Chance", the Galactic Commonwealth is the protagonist of the story, their citizens a mixture of those descendants of manking which managed to flee the end of their world and billions of members of other races who are accepted as citizens. Moreover, due to their focus on reason and research, they are also one of the most advanced realms in their galaxy.
- The Legend of Korra:
- The United Republic of Nations is initially headed by a council made up of each nation's ambassador, which is later replaced by a popularly elected president.
- In the series finale, Prince Wu decides to abolish the Earth Kingdom's monarchy in favor of a more democratic system.