Electroabsorption has been investigated in semi-insulating asymmetric GaAs/AlGaAs double quantum wells presenting high linear Stark responses, adequate for photorefractive applications. We have used the envelope function approximation to calculate the linear Stark shifts of the energy levels and select a suitable structure for the experimental study. The experimental data indicate that the response to the applied field critically depends on a complicated interplay of effects that compete or cooperate to suppress or enhance the electroabsorption. For positive field polarity, the competing contributions of the overlapping $e1$–$hh1$ and $e1$–$hh2$ transitions partially cancel the electroabsorption despite large linear Stark shifts. On the other hand, small negative fields induce large electroabsorption because the Stark shifts of the two transitions have opposite signs.