  We provide the first explicit example of Type IIB string theory
compactification on a globally defined Calabi-Yau threefold with torsion which
results in a four-dimensional effective theory with a non-Abelian discrete
gauge symmetry. Our example is based on a particular Calabi-Yau manifold, the
quotient of a product of three elliptic curves by a fixed point free action of
Z_2 x Z_2. Its cohomology contains torsion classes in various degrees. The main
technical novelty is in determining the multiplicative structure of the
(torsion part of) the cohomology ring, and in particular showing that the cup
product of second cohomology torsion elements goes non-trivially to the fourth
cohomology. This specifies a non-Abelian, Heisenberg-type discrete symmetry
group of the four-dimensional theory.
