Class number divisibility with split real places ($h_A\mid h_B$)

If a finite extension $B/A$ of number fields has a totally ramified finite prime, the ordinary Hilbert class field $H_A$ is disjoint from $B$. Its base change $B{H}_A/B$ is abelian and unramified at finite primes. Its real places also split, since a split-real unramified extension retains this property under base change. It therefore lies in the ordinary Hilbert class field of $B$, proving $h_A\mid h_B$ even when $A$ has real places.