  We consider threefolds that admit a fibration by K3 surfaces over a
nonsingular curve, equipped with a divisorial sheaf that defines a polarisation
of degree two on the general fibre. Under certain assumptions on the threefold
we show that its relative log canonical model exists and can be explicitly
reconstructed from a small set of data determined by the original fibration.
Finally we prove a converse to the above statement: under certain assumptions,
any such set of data determines a threefold that arises as the relative log
canonical model of a threefold admitting a fibration by K3 surfaces of degree
two.
