  We consider compact $CR$ manifolds of arbitrary $CR$ codimension that satisfy
certain geometric conditions in terms of their Levi form. Over these compact
$CR$ manifolds, we construct a deformation of the trivial $CR$ line bundle over
$M$ which is topologically trivial over $M$ but fails to be even locally $CR$
trivializable over any open subset of $M$. In particular, our results apply to
compact Lorentzian $CR$ manifolds of hypersurface type.
