  The two anomalous X-ray pulsars (AXPs) with well-sampled timing histories, 1E
1048.1-5937 and 1E 2259+586, are known to spin down irregularly, with `bumps'
superimposed on an overall linear trend. Here we show that if AXPs are
non-accreting magnetars, i.e. isolated neutron stars with surface magnetic
fields B_0 > 10^{10} T, then they spin down electromagnetically in exactly the
manner observed, due to an effect called `radiative precession'. Internal
hydromagnetic stresses deform the star, creating a fractional difference
epsilon=(I_3-I_1)/I_1 ~ 10^{-8} between the principal moments of inertia I_1
and I_3; the resulting Eulerian precession couples to an oscillating component
of the electromagnetic torque associated with the near-zone radiation fields,
and the star executes an anharmonic wobble with period tau_pr ~ 2 pi / epsilon
Omega(t) ~ 10 yr, where Omega(t) is the rotation frequency as a function of
time t. We solve Euler's equations for a biaxial magnet rotating in vacuo; show
that the computed Omega(t) matches the measured timing histories of 1E
1048.1-5937 and 1E 2259+586; predict Omega(t) for the next 20 years for both
objects; predict a statistical relation between <d(Omega)/dt> and tau_pr, to be
tested as the population of known AXPs grows; and hypothesize that radiative
precession will be observed in future X-ray timing of soft gamma-ray repeaters
(SGRs).
