  Regular triangulations of products of lattice polytopes are constructed with
the additional property that the dual graphs of the triangulations are
bipartite. The (weighted) size difference of this bipartition is a lower bound
for the number of real roots of certain sparse polynomial systems by recent
results of Soprunova and Sottile [Adv. Math. 204(1):116-151, 2006]. Special
attention is paid to the cube case.
