  Experimental and theoretical results are reported for single-photon single
ionization of the tungsten ion W$^{4+}$. Absolute cross sections have been
measured employing the photon-ion merged-beams setup at the Advanced Light
Source in Berkeley. Detailed photon-energy scans were performed at 200~meV
bandwidth in the 40 -- 105~eV range. Theoretical results have been obtained
from a Dirac-Coulomb R-matrix approach employing basis sets of 730 levels for
the photoionization of W$^{4+}$. Calculations were carried out for the
$4f^{14}5s^2 5p^6 5d^2 \; {^3}{\rm F}_{J}$, $J$=2, ground level and the
associated fine-structure levels with $J$=3 and 4 for the W$^{4+}$ ions. In
addition, cross sections have been calculated for the metastable levels
$4f^{14}5s^2 5p^6 5d^2 \; {^3}{\rm P}_{0,1,2},{^1}{\rm D}_{2},{^1}{\rm
G}_{4},{^1}{\rm S}_{0}$. Very satisfying agreement of theory and experiment is
found for the photoionization cross section of W$^{4+}$ which is remarkable
given the complexity of the electronic structure of tungsten ions in low charge
states.
