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Although English historians have defined Reich as being a strictly German concept of sovereign rule, in the German language itself it means "Empire". In English, we speak of the Holy Roman Empire; in German, it is the HRR or "Heiliges Römisches Reich Deutscher Nation".

Wiktionary quotes Busching, who in 1762 explained Reich as a German understanding of "Eminence", from the Latin Regnum, rather than an Empire or a Realm as other cultures would understand it.

One could see it as a word used in English, Russian, Portuguese and many other languages as a designation for any kind of German kingdom, empire or absolutism, similar to how one would address a foreigner according to their origin, e.g. Senor Martinez, Monsieur Aragon.
Compare it to how many cultures, including the Germans, also use the word "Commonwealth" or "Soviet Union", despite the fact that Bavaria had formed its own Union of Soviets, the Räterepublik, during the Weimar Republic.

If you're interested in knowing when the word "Third Reich" came into usage, it is worth noting that during Fascist Germany, the country's official name was Das Deutsche Reich, and in the USA and UK it was usually referred to as Germany, Hitler's Germany, Berlin, Reich or, the most popular term, Nazi Germany.

(Sources: Neville Chamberlain's declaration of war against Germany, headlines about Germany in newspapers like the NY Times or the Times, the Daily Mail's Lord Rothermere)

In Germany itself, the idea of a third Reich stemmed from the 19th century, where it was a popular philosophical theme in literature and theology. Writers like Johannes Schlaf, who wrote "Das Dritte Reich" in 1899, or the German translator of Ibsen's "The Emperor and the Galilean", popularized the idea of the imminent arrival of a thousand years of Christianity (a kind of positive apocalypse), which would follow a third "Reich". And they would quote Paul the Apostle and various saints, to make it clear that the third Reich would follow the heretic lex naturalis and the lex mosaica.

Nazi propagandists exploited this in their articles, making it sound as if the saints had had some sort of epiphany about Hitler and the Nazis.
In other words, the Third Reich began as an abstract idea of a Christian revolution, but was assumed by Nazi writers in order to bolster Hitler's popularity and justify the drastic and bloody decisions that the Fascist government was making.
In one of the earliest examples of the Third Reich being mentioned outside of Germany, Major Strasser in the film Casablanca talks about the Third Reich as if it was just the beginning to a kind of paradisaical future or of a fourth Reich.

After 1945, the term was used exclusively to associate only to the unsuccessful reign of Fascism in Germany, and it ceased as a philosophical or theological idea in German literary circles.

To complement R Mac's answer, Reich entered the English lexicon in this use in the 18th and 19th centuries, so by the time the Third Reich rose in the 1930s, the word would have needed no translation.

Thomas Carlyle, who wrote a history of Friedrich the II of Prussia in 1865, refers to Reich 27 times. The term is applied to both the Holy Roman Empire and to Fredrick the Great's kingdom of Prussia.

The Oxford English Dictionary entry for Reich collects several more examples from 19th century periodicals, principally the Times, including this one from 1852:

Times 6 July 6/4 It was the old court of appeal of the Reich, remarkable in its time, even among other courts, for its majestic slowness of procedure.

So this word would have been readily identified with Germanic centers of power from the medieval period to the present.

Since this word serves principally as a reference, it'd be inaccurate to gloss it as a single word like realm. It's more of a linguistic borrowing, like Khanate, where the form of government and the culture of origin are both bound up within the word. Referring to a non-Mongol/Turkic entity as a Khanate would invite comparison to actual Khanates, just like referring to a non-Germanic entity as a Reich would invite comparison to the HRE/Prussia/Germany (before the 20th century) and to Nazi Germany today.

Interestingly, the "First Reich" is the Holy Roman Empire. So the concept of the "Reich" as understood by speakers of German transcended language and cultural shifts over a very long period of time, from 962 CE through 1945 CE.

You can therefore think of "Reich" as similar to the English "realm" but different. The "Reich" is what German speakers view to be the empire of the German state, with authority derived from a unified "German people" within the empire instead of from a religious deity or a monarchy. As such, a Reich is distinctly German. I assume that since the 1940s every English speaking person on the planet had learned that the word "Reich" essentially means "Realm of Germany", there was never any need to translate it. And translating it would have lost some meaning, since there's no clean way to articulate this concept in English.

I think as to why "Reich" was retained is because of a simple matter of productivity and suitability of the word "Reich;" i.e., the economy of the German word itself warrants retention.

"Third Reich" has contexts:

a) It is related to Germany - because of "Reich"

b) It is related to XX century history - because of "Third"

c) It just sounds "cool"

It's just an economy of words: with "Third Reich" we have a lot of contextual information condensed into just two words.

Comparison with another languages

Russian:
As a native Russian speaker, I can inform you that the same is in Russian language: "Das Dritte Reich" is translated as "Третий Рейх"( "Рейх" is transliteration of "Reich")

I don't know how native English speakers can understand "The Third Realm," but for Russian speaking, "Третий Мир," can be easily confused with concept of Third Rome.

Again, it's merely a question of economy.

Two words clearly defined in their historical and geographical context (and "Reich" sounds cool in Russian - and English - too).

Japanese(and Chinese?): Although if we look further on East we can find that in China and Japan translation 第三帝国 is used (but kanji themselves are embodiment of economy, so my "theory of economy of words" still works)

Swedish: Tredje_riket - looks like translation.

Slovak: Tretia ríša Aha! And in Slovak we have an ambiguity! (Altghough I suppose that for most Slovaks it's not a problem to mix German words with ease).

(Please do not take too seriously my hypothesis. I would be glad to entertain contention if I seem to be incorrect).

The use of the German word "Reich" clearly conveys that one is referring to Germany (or at least to a German-speaking country). If one were to replace "Reich" by "Empire" (or a similar English-language alternative), one would have to refer to "The Third German Empire" (or similar), whereas using the German word "Reich" automatically conveys that it is Germany (or another German-speaking country) that is being referred to, thus removing the need for the extra word.

In other words, The Third Reich is more compact than The Third German Empire", while automatically conveying that it is Germany that is being referred to.

Could not some of the usage be based on the Shirer bestseller “The Rise and Fall of the Third Reich” ?

https://en.wikipedia.org/wiki/The_Rise_and_Fall_of_the_Third_Reich

Not a direct answer, but I'm surprised none of the other comments or answers mentioned that English indeed does have a direct cognate to the German "Reich," as can be seen in the word "Bishopric":

https://www.merriam-webster.com/dictionary/bishopric

This word is still in use in place of "diocese" by various protestant denominations rejecting the latin-derived word in favor of a Germanic one.

So lack of suitable cognates alone can't be the answer. Although it would have been interesting if we had half-translated with the other remaining halves, leaving us with the "dritte ric," rather than "third Reich."

I am going to suggest a rather simplistic reason for this that has little to do with language : space.

In journalism space is money. You (and your editor) will always try and cram as much as possible into the smallest space they can, generally because more free space means more advertising revenue (a little simplistic, but space=money is they idea here). A single letter can push a line of text onto a new printed line, eating into column inches available elsewhere.

An editor faced with "Dritten Reich" will happily accept "Third" as being shorter than Dritten, not to mention that Dritten conveys no information to the English speaking reader. Reich, on the other hand, has no simple translation (and it's clear from the more informed answers here that even agreeing one would be difficult). So those five letters can stay as they are (from the editor's point of view). Not only does "reich" convey the German connection clearly, but the combination (as noted by @Alex-Yu) does sound "cool".

So it may simply boil down to the constant desire to save space in printed news of the day and the soundbite being a good result.

IMHO is is wrong to call Nazi controlled Germany the Third Reich because the Nazis liked to call it the Third Reich, hoping to gain some of the glory associated with that phrase in Germany according to Chris W.'s answer. Of course they never officially changed the name to Third Reich, so Third Reich is also inappropriate because of not being official.

So I think instead that it should be called Germany when describing it as a country and a nation, and Nazi Germany, Nazi controlled Germany, Nazi infested Germany, Nazi contaminated Germany, etc., etc. when describing the Nazi rule of Germany.

Similarly the previous regime should not be called the Weimar Republic or Weimar Germany, because that was a Nazi habit.

The official name of the German state was Deutsches Reich (German Realm/Empire/State/Polity/Country, etc.) from 1871 to 1943, and Grossdeutsches Reich (Greater German Realm/Empire/State/Polity/Country, etc.) from 1943 to 1945. The republic in 1919 took over the bureaucracy and institutions of the previous German Empire of 1871-1918, and the Nazis in turn took over the bureaucracy and institutions of the previous republic in 1933.

There was direct continuity from 1871-1945. So another and third reason not to call Nazi ruled Germany the Third Reich is because Nazi run Germany was actually still the Second Reich. And I am not certain whether, or for how long, or to what degree, the present Federal Republic of Germany was or is considered to be a continuation of the Second Reich. And if the Federal Republic of Germany is not a continuation of the Second Reich that would make it the true Third Reich, and thus calling Nazi Germany the Third Reich would be legitimizing their stealing the term.

In any case the Nazis were jumping the gun by calling their rule the Third Reich.

And what is the English synonym of reich? Realm?

That's why.