With someone that either von Rennenkampf or Samsonov (whoever is left) can work with. Also someone a little more security conscious who transmits in code. Does this make Tannenburg winnable for the Russians?
Not at all... the two army commanders not getting along was only a small portion of the problem. Their army group commander (Jalinsky) and the stavka and the czar where far greater problems.
Due to their committment to France and the akward mobilization schedule the Russians attacked two weeks after the declaration of war (such a rush created several serious problems in East Prussia)
1. Mobile field kitchens were not set up or available (the French had the same problem... you would think in an army of millions of men that feeding them would be a concern... but to the czar and Jalinsky didn't care)
2. Men were not properly outfitted with military footwear (again if you are going to march through the forrests and rough terrain of East Prussia boots might be a good idea but this was an issue at Tanneberg)
3. Codebooks where not distrubted (obvious)
4. Rushed mobilization meant the armies where not on the same marching schedule nor had they been properly briefed
5. Stavka and Jalinsky ordered up attack after attack regardless of enemy action, terrain, hunger/exaustion, or the weather
you would need multiple serious pod's going back to at least 1910 to remedy these issues