  Let $p$ be an odd prime. We give an unconditional proof of the equivariant
Iwasawa main conjecture for totally real fields for every admissible
one-dimensional $p$-adic Lie extension whose Galois group has an abelian Sylow
$p$-subgroup. Crucially, this result does not depend on the vanishing of any
$\mu$-invariant. As applications, we deduce the Coates-Sinnott conjecture away
from its $2$-primary part and new cases of the equivariant Tamagawa number
conjecture for Tate motives.
