  Equivalences between conformal foliations on Euclidean $3$-space, Hermitian
structures on Euclidean $4$-space, shear-free ray congruences on Minkowski
$4$-space, and holomorphic foliations on complex $4$-space are explained
geometrically and twistorially; these are used to show that 1) any
real-analytic complex-valued harmonic morphism without critical points defined
on an open subset of Minkowski space is conformally equivalent to the direction
vector field of a shear-free ray congruence, 2) the boundary values at infinity
of a complex-valued harmonic morphism on hyperbolic $4$-space define a
real-analytic conformal foliation by curves of an open subset of Euclidean
$3$-space and all such foliations arise this way. This gives an explicit method
of finding such foliations; some examples are given.
