  We model the non-thermal transient Swift J1644+57 as resulting from a
relativistic jet powered by the accretion of a tidally-disrupted star onto a
super-massive black hole. Accompanying synchrotron radio emission is produced
by the shock interaction between the jet and the dense circumnuclear medium,
similar to a gamma-ray burst afterglow. An open mystery, however, is the origin
of the late-time radio rebrightening, which occurred well after the peak of the
jetted X-ray emission. Here, we systematically explore several proposed
explanations for this behavior by means of multi-dimensional hydrodynamic
simulations coupled to a self-consistent radiative transfer calculation of the
synchrotron emission. Our main conclusion is that the radio afterglow of Swift
J1644+57 is not naturally explained by a jet with a one-dimensional top-hat
angular structure. However, a more complex angular structure comprised of an
ultra-relativistic core (Lorentz factor $\Gamma \sim 10$) surrounded by a
slower ($\Gamma \sim $ 2) sheath provides a reasonable fit to the data. Such a
geometry could result from the radial structure of the super-Eddington
accretion flow or as the result of jet precession. The total kinetic energy of
the ejecta that we infer of $\sim$ few $10^{53}\,$erg requires a highly
efficient jet launching mechanism. Our jet model providing the best fit to the
light curve of the on-axis event Swift J1644+57 is used to predict the radio
light curves for off-axis viewing angles. Implications for the presence of
relativistic jets from TDEs detected via their thermal disk emission, as well
as the prospects for detecting orphan TDE afterglows with upcoming wide-field
radio surveys and resolving the jet structure with long baseline
interferometry, are discussed.
