= Solution
A ferromagnetic vacuum breaks the <special unitary group> $SU(2)$ to the subgroup of rotations about its magnetization, so its vacuum manifold is the <two-sphere> $S^2$. The massless field is therefore a unit vector $\mathbf n(x,t)$ with $\mathbf n^2=1$.
The only rotational scalar linear in $\dot{\mathbf n}$ that can be formed directly from $\mathbf n$ is $\mathbf n\mathbin\cdot\dot{\mathbf n}=\tfrac12\partial_t(\mathbf n^2)=0$. A local one-form $\mathbf A(\mathbf n)\mathbin\cdot\dot{\mathbf n}$ can describe the required <Berry phase>, but no choice of $\mathbf A$ is globally nonsingular and strictly rotationally invariant on $S^2$. One therefore needs coordinate patches, an extension, or a redundant spinor parametrization.
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