  The radio millisecond pulsar PSR J1023+0038 exhibits complex timing and
eclipse behavior. Here we analyze four years' worth of radio monitoring
observations of this object. We obtain a long-term timing solution, albeit with
large residual timing errors as a result of apparent orbital period variations.
We also observe variable eclipses when the companion passes near our line of
sight, excess dispersion measure near the eclipses and at random orbital
phases, and short-term disappearances of signal at random orbital phases. We
interpret the eclipses as possibly due to material in the companion's
magnetosphere supported by magnetic pressure, and the orbital period variations
as possibly due to a gravitational quadrupole coupling mechanism. Both of these
mechanisms would be the result of magnetic activity in the companion, in
conflict with evolutionary models that predict it should be fully convective
and hence non-magnetic. We also use our timing data to test for orbital and
rotational modulation of the system's $\gamma$-ray emission, finding no
evidence for orbital modulation and $3.7\sigma$ evidence for modulation at the
pulsar period. The energetics of the system make it plausible that the
$\gamma$-ray emission we observe is entirely from the millisecond pulsar
itself, but it seems unlikely for these $\gamma$-rays to provide the
irradiation of the companion, which we attribute instead to X-ray heating from
a shock powered by a particle wind.
