Define socialist republics ?

That depends which country you are in and whether your capitalization of ‘socialist’ and ‘republican’ was intentional or not.

If you were British, or from any other country with a monarchy and you did mean to spell it with lower case letters then it would basically just mean any socialist. All true socialists are republican because they believe that all monarchies should be dismantled and be replaced with socialist republics. The whole idea of a ruling class, rule by the wealthy and rule by birthright is completely anathema to a socialist.

If you did mean to capatalize the letters then it could very well be the name of a political party somewhere. I have no idea if it is true but it would make sense in a country that either has or has had a monarchy and now there is a socialist party who wants or did want to remove the monarch

If you're American then with lower case letters it still makes sense although the word republican is a bit redundant as you live on a republic anyway. If you meant it to have capital letters then it doesn't make any sense because your Republican party is very right wing. A socialist could never be a member of your Republican party. Equally they could never be a member of your Democrats party either, because they are also too right wing (they don't believe in government or co-operative ownership of industry etc).

It should be pointed out that there is nothing inherently right-wing about the word ‘republican’. It just happens that in the USA the right wing party happens to have called themselves the Republican Party. Republican just means “form of government without a monarch” or “person who believes in government without a monarch” while ‘democrat’ just means “person who believes in democracy" and the American centre-right party has used that word as its name. Funnily enough, if you go back in time the Republican Party in the US was more the centrist party while the Democratic Party was more right-wing. They did a bit of a swap about in the early 20th Century

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