  We prove a multivariate strengthening of Brenti's result that every root of
the Eulerian polynomial of type $B$ is real. Our proof combines a refinement of
the descent statistic for signed permutations with the notion of real
stability-a generalization of real-rootedness to polynomials in multiple
variables. The key is that our refined multivariate Eulerian polynomials
satisfy a recurrence given by a stability-preserving linear operator. Our
results extend naturally to colored permutations, and we also give stable
generalizations of recent real-rootedness results due to Dilks, Petersen, and
Stembridge on affine Eulerian polynomials of types $A$ and $C$. Finally,
although we are not able to settle Brenti's real-rootedness conjecture for
Eulerian polynomials of type $D$, nor prove a companion conjecture of Dilks,
Petersen, and Stembridge for affine Eulerian polynomials of types $B$ and $D$,
we indicate some methods of attack and pose some related open problems.
