Context. DG Tau is a low-mass pre-main sequence star, whose strongly accreting protoplanetary disk exhibits a so-far enigmatic behavior: its mid-infrared thermal emission is strongly time-variable, even turning the 10 $mu$m silicate feature from emission to absorption temporarily. Aims. We look for the reason for the spectral variability at high spatial resolution and at multiple epochs. Methods. We study the temporal variability of the mid-infrared interferometric signal, observed with the VLTI/MIDI instrument at six epochs between 2011 and 2014. We fit a geometric disk model to the observed interferometric signal to obtain spatial information about the disk. We also model the mid-infrared spectra by template fitting to characterize the profile and time dependence of the silicate emission. We use physically motivated radiative transfer modeling to interpret the mid-infrared interferometric spectra. Results. The inner disk (r<1-3 au) spectra exhibit a 10 $mu$m absorption feature related to amorphous silicate grains. The outer disk (r>1-3 au) spectra show a crystalline silicate feature in emission, similar to the spectra of comet Hale-Bopp. The striking difference between the inner and outer disk spectral feature is highly unusual among T Tauri stars. The mid-infrared variability is dominated by the outer disk. The strength of the silicate feature changed by more than a factor of two. Between 2011 and 2014 the half-light radius of the mid-infrared-emitting region decreased from 1.15 to 0.7 au. Conclusions. For the origin of the absorption we discuss four possible explanations: a cold obscuring envelope, an accretion heated inner disk, a temperature inversion on the disk surface and a misaligned inner geometry. The silicate emission in the outer disk can be explained by dusty material high above the disk plane, whose mass can change with time, possibly due to turbulence in the disk.
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