  Periodically driven quantum many-body systems support anomalous topological
phases of matter, which cannot be realized by static systems. In many cases,
these anomalous phases can be many-body localized, which implies that they are
stable and do not heat up as a result of the driving. What types of anomalous
topological phenomena can be stabilized in driven systems, and in particular,
can an anomalous phase exhibiting non-Abelian anyons be stabilized? We address
this question using an exactly solvable, stroboscopically driven 2D Kitaev spin
model, in which anisotropic exchange couplings are boosted at consecutive time
intervals. The model shows a rich phase diagram which contains anomalous
topological phases. We characterize these phases using weak and strong
scattering-matrix invariants defined for the fermionic degrees of freedom. Of
particular importance is an anomalous phase whose zero flux sector exhibits
fermionic bands with zero Chern numbers, while a vortex binds a pair of
Majorana modes, which as we show support non-Abelian braiding statistics. We
further show that upon adding disorder, the zero flux sector of the model
becomes localized. However, the model does not remain localized for a finite
density of vortices. Hybridization of Majorana modes bound to vortices form
"vortex bands", which delocalize by either forming Chern bands or a thermal
metal phase. We conclude that while the model cannot be many-body localized, it
may still exhibit long thermalization times, owing to the necessity to create a
finite density of vortices for delocalization to occur.
