Global tomographic models collected in the Seismic wave Propagation and Imaging in Complex (SPICE media: a European network) model library (http://www.spice-rtn.org) share a similar pattern of long, spatial wavelength heterogeneity, but are not consistent at shorter spatial wavelengths. Here, we assess the performance of global tomographic models by comparing how well they fit seismic waveform observations, in particular Love and Rayleigh wave overtones and fundamental modes. We first used the coupled spectral element method (CSEM) to calculate long-period (> 100 s) synthetic seismograms for different global tomography models. The CSEM can incorporate the effect of three-dimensional (3-D) variations in velocity, anisotropy, density and attenuation with very little numerical dispersion. We then compared quantitatively synthetic seismograms and real data. To restrict ourselves to high-quality overtone data, and to minimize the effects of the finite extent of seismic sources and of crustal heterogeneity, we favour deep (> 500 km) earthquakes of intermediate magnitude (M-w similar to 7). Our comparisons reveal that: (1) The 3-D global tomographic models explain the data much better than the one-dimensional (1-D) anisotropic Preliminary Reference Earth Model (PREM). The current 3-D tomographic models have captured the large-scale features of upper-mantle heterogeneities, but there is still some room for the improvement of large-scale features of global tomographic models. (2) The average correlation coefficients for deep events are higher than those for shallow events, because crustal structure is too complex to be completely incorporated into CSEM simulations. (3) The average correlation coefficient (or the time lag) for the major-arc wave trains is lower (or higher) than that for the minor-arc wave trains. Therefore, the current tomographic models could be much improved by including the major-arc wave trains in the inversion. (4) The shallow-layer crustal correction has more effects on the fundamental surface waves than on the overtones.