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The recently detected extended very-high-energy (VHE) γ-ray emission from the microquasar V4641 Sgr which reaches 800 TeV, has led to significant speculation about the nature of the accelerator. The VHE emission reveals a puzzling 200 parsec-long jet-like structure substantially misaligned with the microquasar’s radio jet which is observed in blazar mode. In [1], we proposed that this γ-ray structure is produced by high-energy cosmic-ray particles escaping from the microquasar along ordered field lines of the Galactic Magnetic Field and interacting with the interstellar medium. In this contribution, we summarise the results of [1] and extend the study of V4641 Sgr. We model the multiwavelength spectral energy distribution with leptonic, hadronic, and mixed leptohadronic models. By comparing these three broad classes of models, we find that a leptohadronic origin best accounts for the spectral characteristics and the morphological features observed in the nebula simultaneously. Finally, we model the expected neutrino flux from V4641 Sgr and show the system is detectable by future multi-km3 neutrino detectors.