I've heard this a few times and never thought much of it. But now I'm curious, because it certainly sounds like it could be a myth, and when I looked into a bit, it seems there aren't much in the way of real sources besides unsubstantiated account of Tom Standage.
Do we know if Coke really secretly made White Coke for the Soviets, or is it a myth?
I'm assuming you've looked at the Wiki page for this, so I won't bother pointing you there.
The only vaugley scholarly source that addresses this in any detail is Mark Pendergast's For God, Country, and Coca-Cola. Basically, Pendergast concludes that the "white coke" story is at least plausible, and very even probable. When the book was written in 1993, Pendergast intervied Mladin Zarubica, an American-raised engineer who had worked for Coca-Cola in Austria after the war, and Zarubica claimed the story was true, but that it was only one of several wild-and-wooly schemes that Coca-Cola was using at the time to try to gain market share in Central Europe.
Pendergast's book is quite positive toward the company, but on the other hand, it seems quite well-researched, so if the story is false, it's a falsehood that's within the realm of possibility. The Coca-Cola Company has always been very aggressive in trying to open new markets, so it's not beyond the realm of possibility that the would authorize one of their European subsidiaries to produce such a product if they thought it might help them open up otherwise closed markets in Eastern Europe.