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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 $ν$. 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.