  In many-cycle plane waves at intermediate intensities, the nonlinear trident
process can be well-approximated by the two sequential steps of nonlinear
Compton scattering of a polarised real photon followed by its transformation
into an electron-positron pair via nonlinear Breit-Wheeler pair creation. We
investigate this two-step process in the intermediate intensity regime by
employing the locally monochromatic approximation for each step and numerically
evaluating resulting expressions. When photon polarisation is included, it is
found to produce an order 10% decrease in the trident rate: the importance of
polarisation increases at lower intensities, and decreases at higher
intensities. Its importance persists at higher intensities in a
linearly-polarised background, but disappears at high intensities in a
circularly-polarised background. If the two steps are made to take place in two
linearly-polarised plane wave pulses with perpendicular polarisations, the pair
yield can be increased by approximately 30% compared to two plane waves with
the same polarisation. It is also shown that harmonic structures in the Compton
step can be passed to the pair step if the Compton edge is at an energy of the
order of the threshold for linear Breit-Wheeler.
