  Double parton scattering (DPS) processes in which there is a perturbative
$1\to2$ splitting in both protons overlap with loop corrections to single
parton scattering (SPS). Any fundamental theoretical treatment of DPS needs to
address this double-counting issue. In this paper, we augment our Monte-Carlo
simulation of DPS, dShower, to be able to generate kinematic distributions
corresponding to the combination SPS+DPS without double counting. To achieve
this, we formulate a fully-differential version of the subtraction scheme
introduced in Diehl et al. (JHEP 06 (2017) 083). A shower is attached to the
subtraction term, and this is combined with the dShower DPS shower along with
the usual SPS shower. We perform a proof-of-concept study of this new algorithm
in the context of $\mathrm{Z}^0\mathrm{Z}^0$ production. Once the subtraction
term is included, we verify that the results do not depend strongly on the
artificial "DPS-SPS demarcation" scale $\nu$. As part of the development of the
new algorithm, we improve the kinematics of the $1\to2$ splitting in the DPS
shower (and subtraction term), allowing the daughter partons to have a relative
transverse momentum. Several reasonable choices for the transverse profile in
the $1\to2$ splitting are studied. We find that many kinematic distributions
are not strongly affected by the choice, although we do observe some
differences in the region where the transverse momenta of both bosons are
small.
