  Let $A/K$ be an abelian variety with real multiplication defined over a
$p$-adic field $K$ with $p>2$. We show that $A/K$ must have either potentially
good or potentially totally toric reduction. In the former case we give
formulas of the local root number of $A/K$ under the condition that inertia
acts via an abelian quotient on the associated Tate module; in the latter we
produce formulas without additional hypotheses.
