So what exactly happens to the real interest rate in the long run?
Blanchard examines a case with a single interest rate, and what happens to it "in the medium long-run" (in the sense of the economy being in equilibrium). His discussion is the most abstract. 

Acemoglou discusses a more realistic case where many different interest rates exist at the same time, and analyzes the observed phenomenon that interest rates on short-term loans are different than interest rates on long-term loans, even when both contracts are entered into at the same calendar time: the time horizon of the contract matters for the interest rate itself. He then goes on to discuss how monetary policy may or may not affect expectations. 

So when Acemoglou says "the long-term real interest rate tends to fall..." he does not mean "the real interest rate characterizing long-run equilibrium tends to fall". He means "the real interest rate that is currently imposed on loans with a long-term horizon, tends to fall". 

These aspects are simply absent in the basic IS-LM / AS-AD analysis.