Efficiently simulating light transport in various scenes with a single algorithm is a difficult and important problem in computer graphics. Two major issues have been shown to hinder the efficiency of the existing solutions: light transport due to multiple highly glossy or specular interactions, and scenes with complex visibility between the camera and light sources. While recent bidirectional path sampling methods such as vertex connection and merging/unified path sampling (VCM/UPS) efficiently deal with highly glossy or specular transport, they tend to perform poorly in scenes with complex visibility. On the other hand, Markov chain Monte Carlo (MCMC) methods have been able to show some excellent results in scenes with complex visibility, but they behave unpredictably in scenes with glossy or specular surfaces due to their fundamental issue of sample correlation. In this paper, we show how to fuse the underlying key ideas behind VCM/UPS and MCMC into a single, efficient light transport solution. Our algorithm is specifically designed to retain the advantages of both approaches, while alleviating their limitations. Our experiments show that the algorithm can efficiently render scenes with both highly glossy or specular materials and complex visibility, without compromising the performance in simpler cases.