Thresholds were measured for detecting sinusoidal amplitude modulation in the presence of a complex-tone masker modulation. Both modulations were applied to the same sinusoidal carrier. Two different masker modulations were used: (i) a pair of components beating at the difference frequency and (ii) a three-tone complex producing a sinusoidal amplitude modulation of the modulation depth at the difference frequency between adjacent components. Both maskers show a periodicity in the waveform that is not contained in the envelope spectrum itself but can be observed when the envelope of the envelope, referred to as the "venelope" [Ewert et al., J. Acoust. Soc. Am. 112. 2921-2931 (2002)], is calculated. For a signal frequency equal to the masker-venelope periodicity, modulation depth at threshold was measured as a function of the signal phase relative to the phase of the masker-venelope component. Signal frequencies of 5, 30, and 90 Hz were used. It was found that masking was phase dependent for all three signal frequencies. Thresholds were lower for the in-phase condition, where maxima in the signal waveform coincided with maxima in the masker-venelope waveform, than for the antiphase condition. The maximum threshold difference was 15 dB. The results are in contrast to recent data [Moore et al., J. Acoust. Soc. Am. 106, 908-918 (1999)], where lowest thresholds were reported for the antiphase condition in a similar experiment. The present data are in line with the idea that a nonlinearity prior to a modulation filterbank extracts the venelope of the masker modulator. However, a compressive nonlinearity such as that associated with the processing on the basilar membrane cannot account for the empirical findings, since it predicts the opposite phase effect.