  A large fraction of top quark events in $\pp$ collisions at $1.8\ \TeV$ will
contain additional soft hadronic jets from gluon bremsstrahlung off the quarks
and gluons in the hard processes $\qq, gg \to \tt \to \bb \ww$. These extra
jets can cause complications when attempting to reconstruct $m_t$ from the
invariant mass of combinations of final-state quarks and leptons. We show how
such soft radiation cannot be unambiguously associated with either
initial-state radiation or or with final-state radiation off the $b$ quarks.
The top quarks can radiate too, and in fact the pattern of radiation has a very
rich structure, which depends on the orientation of the final-state particles
with respect to each other and with respect to the beam. We calculate the full
radiation pattern of soft jets in the soft gluon approximation and compare with
several approximate forms which are characteristic of parton shower Monte
Carlos. The implications for top mass measurements are discussed.
