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We performed a full mapping of the bulk electronic structure including the Fermi surface and Fermi-velocity distribution v_F(k_F) of tungsten. The 4D spectral function ρ(E_B; k) in the entire bulk Brillouin zone and 6 eV binding-energy (E_B) interval was acquired in ∼3 h thanks to a new multidimensional photoemission data-recording technique (combining full-field k-microscopy with time-of-flight parallel energy recording) and the high brilliance of the soft X-rays used. A direct comparison of bulk and surface spectral functions (taken at low photon energies) reveals a time-reversal-invariant surface state in a local bandgap in the (110)-projected bulk band structure. The surface state connects hole and electron pockets that would otherwise be separated by an indirect local bandgap. We confirmed its Dirac-like spin texture by spin-filtered momentum imaging. The measured 4D data array enables extraction of the 3D …