Understanding speech in background noise is a difficult task. The tracking of speech rhythms such as the rate of syllables and words by cortical activity has emerged as a key neural mechanism for speech in-noise comprehension. In particular, recent investigations have used transcranial alternating current stimulation (tACS) with the envelope of a speech signal to influence the cortical speech tracking, demonstrating that this type of stimulation modulates comprehension and therefore evidencing a functional role of the cortical tracking in speech processing. Cortical activity has been found to track the rhythms of a background speaker as well, but the functional significance of this neural response remains unclear. Here we employ a speech-comprehension task with a target speaker in the presence of a distractor voice to show that tACS with the speech envelope of the target voice as well as tACS with the envelope of the distractor speaker both modulate the comprehension of the target speech. Because the envelope of the distractor speech does not carry information about the target speech stream, the modulation of speech comprehension through tACS with this envelope evidences that the cortical tracking of the background speaker affects the comprehension of the foreground speech signal. The phase dependency of the resulting modulation of speech comprehension is, however, opposite to that obtained from tACS with the envelope of the target speech signal. This suggests that the cortical tracking of the ignored speech stream and that of the attended speech stream may compete for neural resources.