  We describe a class of compact $G_2$ orbifolds constructed from
non-symplectic involutions of K3 surfaces. Within this class, we identify a
model for which there are infinitely many associative submanifolds contributing
to the effective superpotential of M-theory compactifications. Under a chain of
dualities, these can be mapped to F-theory on a Calabi-Yau fourfold, and we
find that they are dual to an example studied by Donagi, Grassi and Witten.
Finally, we give two different descriptions of our main example and the
associative submanifolds as a twisted connected sum.
