  We study the $S=1/2$ pyrochlore Heisenberg antiferromagnet in a magnetic
field. Using large scale density-matrix renormalization group (DMRG)
calculations for clusters up to $128$ spins, we find indications for a finite
triplet gap, causing a threshold field to nonzero magnetization in the
magnetization curve. We obtain a robust saturation field consistent with a
magnon crystal, although the corresponding $5/6$ magnetization plateau is very
slim and possibly unstable. Most remarkably, there is a pronounced and
apparently robust $\frac 1 2$ magnetization plateau where the groundstate
breaks (real-space) rotational symmetry, exhibiting oppositely polarised spins
on alternating kagome and triangular planes. Reminiscent of the kagome ice
plateau of the pyrochlore Ising magnet known as spin ice, it arises via a much
more subtle `quantum order by disorder' mechanism.
