Real rank of some multiplier algebras

We show that there exists a separable, nuclear C*-algebra with real rank zero and trivial K-theory such that its multiplier and corona algebra have real rank one. This disproves two conjectures of Brown and Pedersen. We also compute the real rank of the stable multiplier algebra and the stable corona algebra of countably decomposable type $\mathrm{I}_\infty$ and type $\mathrm{II}_\infty$ factors. Together with results of Zhang this completes the computation of the real rank for stable multiplier and corona algebras of countably decomposable factors.