Antithetic coupling is a general stratification strategy for reducing Monte Carlo variance without increasing the simulation size. The use of the antithetic principle in the Monte Carlo literature typically employs two strata via antithetic quantile coupling. We demonstrate here that further stratification, obtained by using k>2 (e.g., k=3–10) antithetically coupled variates, can offer substantial additional gain in Monte Carlo efficiency, in terms of both variance and bias. The reason for reduced bias is that antithetically coupled chains can provide a more dispersed search of the state space than multiple independent chains. The emerging area of perfect simulation provides a perfect setting for implementing the k-process parallel antithetic coupling for MCMC because, without antithetic coupling, this class of methods delivers genuine independent draws. Furthermore, antithetic backward coupling provides a very convenient theoretical tool for investigating antithetic forward coupling. However, the generation of k>2 antithetic variates that are negatively associated, that is, they preserve negative correlation under monotone transformations, and extremely antithetic, that is, they are as negatively correlated as possible, is more complicated compared to the case with k=2. In this paper, we establish a theoretical framework for investigating such issues. Among the generating methods that we compare, Latin hypercube sampling and its iterative extension appear to be general-purpose choices, making another direct link between Monte Carlo and quasi Monte Carlo.