  In a recent paper, here referred to as part I, we considered the celestial
four-gluon amplitude with one gluon represented by the shadow transform of the
corresponding primary field operator. This correlator is ill-defined because it
contains branch points related to the presence of conformal blocks with complex
spin. In this work, we adopt a procedure similar to minimal models and
construct a single-valued completion of the shadow correlator, in the limit
when the shadow is "soft." By following the approach of Dotsenko and Fateev, we
obtain an integral representation of such a single-valued correlator. This
allows inverting the shadow transform and constructing a single-valued
celestial four-gluon amplitude. This amplitude is drastically different from
the original Mellin amplitude. It is defined over the entire complex plane and
has correct crossing symmetry, OPE and bootstrap properties. It agrees with all
known OPEs of celestial gluon operators. The conformal block spectrum consists
of primary fields with dimensions $\Delta=m+i \lambda$, with integer $m\geq 1$
and various, but always integer spin, in all group representations contained in
the product of two adjoint representations.
