Well, they claimed to have socialism and to be building communism, but Marx himself would hardly approve or consider it "socialism" if comparing notes. There's a reason why Lenin and company were seen as hacks by many Marxists of their time.

But I don't think Walter had that degree of precision in mind when he said "Marxist". It sounded more like a generic label for anti-capitalist left to me, so I went with the broad definition to match that.