Unsupervised foreground-background segmentation aims at extracting salient objects from cluttered backgrounds, where Generative Adversarial Network (GAN) approaches, especially layered GANs, show great promise. However, without human annotations, they are typically prone to produce foreground and background layers with non-negligible semantic and visual confusion, dubbed "information leakage", resulting in notable degeneration of the generated segmentation mask. To alleviate this issue, we propose a simple-yet-effective explicit layer independence modeling approach, termed Independent Layer Synthesis GAN (ILSGAN), pursuing independent foreground-background layer generation by encouraging their discrepancy. Specifically, it targets minimizing the mutual information between visible and invisible regions of the foreground and background to spur interlayer independence. Through in-depth theoretical and experimental analyses, we justify that explicit layer independence modeling is critical to suppressing information leakage and contributes to impressive segmentation performance gains. Also, our ILSGAN achieves strong state-of-the-art generation quality and segmentation performance on complex real-world data.