  The motivation for this work stems from the authors' desire to revive the
study of ``fat bundles''. Recently, several new works have emerged in this
field, including \cite{DV2022, cavenaghilinollohann2, ovando1, ovando2,
BOCHENSKI2016131}, while the most notable reference being W. Ziller's
unpublished notes on fat bundles (\cite{Ziller_fatnessrevisited}). Given these
developments, it is natural to revisit this topic, recalling known facts and
furnishing a new perspective to the known results, despite obtaining new ones.
  One uses recent advances in positive curvatures to obtain a simple and
geometric classification of fat homogeneous Riemannian foliations. Although
B\'erard-Bergery established a similar classification in \cite{bergeryfat}, we
believe one of the novelties of this paper lies in the combination of dual
foliations and Ambrose--Singer's Theorem with metric deformations: motivated by
the regularization properties of Cheeger deformations and the non-metric
dependence of fatness, we show that: the vertical bundle of a fat submersion is
obstructed, being wholly generated by the O'Neill tensor (this is a consequence
of a dual-leaf argument); we prove that any fat foliation induced by the fibers
of a fiber bundle with compact structure group is \emph{twisted} (any
Riemannian submersion compatible metric has dual-foliation with only one dual
leaf). We also prove that if the total space of a fat Riemannian submersion has
non-negative sectional curvature, then their leaves are a symmetric space. We
conclude further classification results.
