Traditionally, exactness, numerical stability and speed are the three main criteria for evaluating algorithms for random variate generation. However, it is sometimes required that the algorithms provide correlation between generated variates for the purpose of inducing dependence among the output of simulation runs. The inverse transformation, which produces optimal correlation induction, often performs poorly in terms of the first three criteria. Algorithms based on composition, rejection, and special properties which often excel in terms of the first three criteria, tend to scramble the use of random numbers, causing many attempts at common random numbers, antithetic variates and external control variates to fail. The concept of obtaining correlation via algorithms other than the inverse transformation is examined here. To demonstrate feasibility, previously developed algorithms for Poisson and binomial random variate generation are modified to obtain both positive and negative correlation between runs....