Speech recognition in adverse real-world environments is highly affected by reverberation and nonstationary background noise. A well-known strategy to reduce such undesired signal components in multi-microphone scenarios is spatial filtering of the microphone signals. In this article, we demonstrate that an additional coherence-based postfilter, which is applied to the beamformer output signal to remove diffuse interference components from the latter, is an effective means to further improve the recognition accuracy of modern deep learning speech recognition systems. To this end, the recently updated 3rd CHiME Speech Separation and Recognition Challenge (CHiME-3) baseline speech recognition system is extended by a coherence-based postfilter and the postfilter's impact on the word error rates is investigated for the noisy environments provided by CHiME-3. To determine the time- and frequency-dependent postfilter gains, we use a Direction-of-Arrival (DOA)-dependent and a DOA-independent estimator of the coherent-to-diffuse power ratio as an approximation of the short-time signal-to-noise ratio. Our experiments show that incorporating coherence-based postfiltering into the CHiME-3 baseline speech recognition system leads to a significant reduction of the word error rate scores for the noisy and reverberant environments provided as part of CHiME-3.