It is well known that trivariate reduction - a method to generate two dependent random variables from three independent random variables - can be used to generate Poisson random variables with specified marginal distributions and correlation structure. The method, however, works only for positive correlations. Moreover, the proportion of feasible positive correlations that can be generated through trivariate reduction deteriorates rapidly as the discrepancy between the means of the target marginal distributions increases. We present a specialized algorithm for generating Poisson random vectors, through appropriate modifications to trivariate reduction. The proposed algorithm covers the entire range of feasible correlations in two dimensions, and preliminary tests have demonstrated very fast preprocessing and generation times.