  In a transparent three-terminal Josephson junction, modeling nonequilibrium
transport is numerically challenging, owing to the interplay between multiple
Andreev reflection (MAR) thresholds and multipair resonances in the pair
current. An approximate method, coined as "partially resummed perturbation
theory in the number of nonlocal Green's functions", is presented that can be
operational on a standard computer and demonstrates compatibility with results
existing in the literature. In a linear structure made of two neighboring
interfaces (with intermediate transparency) connected by a central
superconductor, tunneling through each of the interfaces separately is taken
into account to all orders. On the contrary, nonlocal processes connecting the
two interfaces are accounted for at the lowest relevant order. This yields
logarithmically divergent contributions at the gap edges, which are sufficient
as a semi-quantitative description. The method is able to describe the current
in the full two-dimensional voltage range, including commensurate as well as
incommensurate values. The results found for the multipair (for instance
quartet) current-phase characteristics as well as the MAR thresholds are
compatible with previous results. At intermediate transparency, the multipair
critical current is much larger than the background MAR current, which supports
an experimental observation of the quartet and multipair resonances. The paper
provides a proof of principle for addressing in the future the interplay
between quasiparticles and multipairs in four-terminal structures.
