  In this paper, we characterize all links in the 3-sphere with bridge number
at least three that have a bridge sphere of distance two. We show that a link L
has a bridge sphere of distance at most two then it falls into at least one of
three categories:
  (1) The exterior of L contains an essential meridional sphere.
  (2) L can be decomposed as a tangle product of a Montesinos tangle with an
essential tangle in a way that respects the bridge surface and either the
Montesinos tangle is rational or the essential tangle contains an
incompressible, boundary-incompressible annulus.
  (3) L is obtained by banding from another link L' that has a bridge sphere of
the same Euler characteristic as the bridge sphere for L but of distance 0 or
1.
