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Well, I only have impressionary evidence on the matter, but I am 100% that two Soviet officers put in a recommendation for an Order of Lenin to be awarded to Guido Picelli, a famous Italian international brigadist. Which would implicitly certify that yes, it would have been technically possible for Soviet decorations to be awarded to Republican fighters..
(Picelli, former communist MP, organised the only armed resistance to fascism in Italy, won the "battle of Parma" [Oltretorrente] against the fascists, was later imprisoned and then emigrated to the USSR. He seems to have become disillusioned there and he travelled to Spain to fight for the Republic. He got close to the Libertarians of POUM but was then convinced to join in the Italian Int. Brigade. He died heroically in battle. Some say. Other say he was shot at the back, presumably under Soviet orders, when the NKVD started going after Anarchists and Libertarians in the Republican ranks - research still hasn't entirely cleared up what happened there).
(Picelli, former communist MP, organised the only armed resistance to fascism in Italy, won the "battle of Parma" [Oltretorrente] against the fascists, was later imprisoned and then emigrated to the USSR. He seems to have become disillusioned there and he travelled to Spain to fight for the Republic. He got close to the Libertarians of POUM but was then convinced to join in the Italian Int. Brigade. He died heroically in battle. Some say. Other say he was shot at the back, presumably under Soviet orders, when the NKVD started going after Anarchists and Libertarians in the Republican ranks - research still hasn't entirely cleared up what happened there).
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Very interesting what you are saying. I am researching among Spanish. If I can get information, I'll post it here.
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Soviet union issued commemorative badges well after the events, in 1969... but I'm unsure soviet soldiers even had orders or medals granted for training spanish troops. Maybe badges for NKVD personnal? We need evidences from individual accounts, and I believe they were scarce.