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We give the first examples of $\mathcal{O}$-acyclic smooth projectivegeometrically connected varieties over the function field of a complex curve,whose index is not equal to one. More precisely, we construct a family ofEnriques surfaces over $\mathbb{P}^{1}$ such that any multi-section has evendegree over the base $\mathbb{P}^{1}$ and show moreover that we can find such afamily defined over $\mathbb{Q}$. This answers affirmatively a question ofColliot-Th\'el\`ene and Voisin. Furthermore, our construction providescounterexamples to: the failure of the Hasse principle accounted for by thereciprocity obstruction; the integral Hodge conjecture; and universality ofAbel-Jacobi maps.

An $\mathcal{O}$-acyclic variety of even index