  We address a variant of Zariski Cancellation Problem, asking whether two
varieties which become isomorphic after taking their product with an algebraic
torus are isomorphic themselves. Such cancellation property is easily checked
for curves, is known to hold for smooth varieties of log-general type by virtue
of a result of Iitaka-Fujita, and more generally for varieties which are not
dominantly covered by images of the punctured affine line. We show in contrast
that cancellation fails in general in every dimension bigger or equal to two.
