  We study a model of a thermoelectric nanojunction driven by
vibrationally-assisted tunneling. We apply the reaction coordinate formalism to
derive a master equation governing its thermoelectric performance beyond the
weak electron-vibrational coupling limit. Employing full counting statistics we
calculate the current flow, thermopower, associated noise, and efficiency
without resorting to the weak vibrational coupling approximation. We
demonstrate intricacies of the power-efficiency-precision trade-off at strong
coupling, showing that the three cannot be maximised simultaneously in our
model. Finally, we emphasise the importance of capturing non-additivity when
considering strong coupling and multiple environments, demonstrating that an
additive treatment of the environments can violate the upper bound on
thermoelectric efficiency imposed by Carnot.

Quantum Physics;Mesoscale and Nanoscale Physics