  We study Q-factorial terminal Fano 3-folds whose equations are modelled on
those of the Segre embedding of P^2 x P^2. These lie in codimension 4 in their
total anticanonical embedding and have Picard rank 2. They fit into the current
state of classification in three different ways. Some families arise as
unprojections of degenerations of complete intersections, where the generic
unprojection is a known prime Fano 3-fold in codimension 3; these are new, and
an analysis of their Gorenstein projections reveals yet other new families.
Others represent the "second Tom" unprojection families already known in
codimension 4, and we show that every such family contains one of our models.
Yet others have no easy Gorenstein projection analysis at all, so prove the
existence of Fano components on their Hilbert scheme.
