  This paper is devoted to the study of a certain class of principal bundles on
del Pezzo surfaces, which were introduced and studied by Friedman and Morgan in
\cite{FMdP}: The two authors showed that there exists a unique principal bundle
(up to isomorphism) on a given (Gorenstein) del Pezzo surface satisfying
certain properties. We call these bundles {\em almost regular}. In turn, we
study them in families. In this case, the existence and the moduli of these
bundles are governed by the cohomology groups of an abelian sheaf ${\mathscr
A}$: On a given del Pezzo fibration, the existence of an almost regular bundle
depends on the vanishing of an obstruction class in $H^2({\mathscr A})$. In
which case, the set of isomorphism classes of almost regular bundles become a
homogeneous space under the $H^1({\mathscr A})$ action.
