The complex binary systemβLyr A has an extensive observational dataset: light curves (from far UV to far IR), interferometric squared visibility, closure phase, triple product measurements, spectral-energy distribution, high-resolution spectroscopy, differential visibility amplitude, and also a differential phase. In particular, we used spectra from the Ondřejov 2m telescope from 2013 to 2015 to measure the emission in Hα, He I, Si II, Ne I, or C IIlines, and differential interferometry by CHARA/VEGA from the 2013 campaign to measure wavelength-dependent sizes across Hαand He I6678. This allowed us to constrain not only optically thick objects (primary, secondary, accretion disc), but also optically thin objects (disc atmosphere, jets, shell). We extended our modelling tool, Pyshellspec (based on Shellspec; a 1D local thermodynamical equilibrium radiative transfer code), to include all new observables, to compute differential visibilities/phases, to perform a Doppler tomography, and to determine a jointχ²metric. After an optimisation of 38 free parameters, we derived a robust model of theβLyr A system. According to the model, the emission is formed in an extended atmosphere of the disc, two perpendicular jets expanding at ∼700 km s⁻¹, and a symmetric shell with the radius ∼70 R_⊙. The spectroscopy indicates a low abundance of carbon, 10⁻²of the solar value. We also quantified systematic differences between datasets, and we discuss here alternative models with higher resolutions, additional asymmetries, or He-rich abundances.