  We consider the plus-phase of the two-dimensional Ising model below the
critical temperature. In $1989$ Schonmann proved that the projection of this
measure onto a one-dimensional line is not a Gibbs measure. After many years of
continued research which have revealed further properties of this measure, the
question whether or not it is a Gibbs measure in an almost sure sense remains
open.
  In this paper we study the same measure by interpreting it as a temporal
process. One of our main results is that the Schonmann projection is almost
surely a regular $g$-measure. That is, it does possess the corresponding
one-sided notion of almost Gibbsianness. We further deduce strong one-sided
mixing properties which are of independent interest. Our proofs make use of
classical coupling techniques and some monotonicity properties which are known
to hold for one-sided, but not two-sided conditioning for FKG measures.
