  Let $K$ be a cyclic totally real number field of odd degree over $\mathbb{Q}$
with odd class number, such that every totally positive unit is the square of a
unit, and such that $2$ is inert in $K/\mathbb{Q}$. We define a family of
number fields $\{K(p)\}_p$, depending on $K$ and indexed by the rational primes
$p$ that split completely in $K/\mathbb{Q}$, such that $p$ is always ramified
in $K(p)$ of degree $2$. Conditional on a standard conjecture on short
character sums, the density of such rational primes $p$ that exhibit one of two
possible ramified factorizations in $K(p)/\mathbb{Q}$ is strictly between $0$
and $1$ and is given explicitly as a formula in terms of $[K:\mathbb{Q}]$. Our
results are unconditional in the cubic case. Our proof relies on a detailed
study of the joint distribution of spins of prime ideals.
