Understanding why Berry phase, as you parallel transport along the geodesic is not zero

Parallel transporting a state along a geodesic doesn't introduce any anholonomy angle, that's what I learned in general relativity. In quantum mechanics, this anholonomy for states are related to the Berry phase. As one takes the state $$\begin{pmatrix}\cos(\frac{\theta}{2}) \\ \sin(\frac{\theta}{2})e^{i \phi} \end{pmatrix}$$ And performs a parallel transport along the equator of the Bloch sphere (or along any other great circle for that matter) on obtains a $$-\pi$$ Berry phase. But I thought there was no anholonomy when you go along a geodesic, here along the great circles of the Bloch sphere.

The $$-\pi$$ Berry phase makes sense if we see it from the other viewpoint that an electronic wave function gets a $$\exp(-i \pi)$$ as it rotates $$2\pi$$. What I'm missing about the parallel transport in this case?