  We show that for each fixed dimension $d\geq 2$, the set of $d$-dimensional
klt elliptic varieties with numerically trivial canonical bundle is bounded up
to isomorphism in codimension one, provided that the torsion index of the
canonical class is bounded and the elliptic fibration admits a rational
section. This case builds on an analogous boundedness result for the set of
rationally connected log Calabi-Yau pairs with bounded torsion index. In
dimension $3$, we prove the more general statement that the set of
$\epsilon$-lc pairs $(X,B)$ with $-(K_X +B)$ nef and rationally connected $X$
is bounded up to isomorphism in codimension one.
