For this question, a Calabi-Yau manifold or variety of dimension $n$ is defined as a non-singular projective variety with trivial canonical bundle and $h^{i,0} = 0$ unless $i = 0$ or $i = n$.

If there are no mistakes in this argument, the base of an elliptically fibered Calabi-Yau threefold $\pi:X\to B$ has to be rational if $X$ has a trivial fundamental group, but if the fundamental group is not trivial, the base $B$ might be an Enriques surface.

Is it actually possible for the base to be an Enriques surface? Are there restrictions on the fundamental group of the Calabi-Yau variety in that case?