We define an algorithm on fermionic and bosonic twisted multiline queues that projects to the multispecies totally asymmetric simple exclusion process (TASEP) and the totally asymmetric zero range process (TAZRP) on a ring, respectively. Our algorithm on fermionic multiline queues generalizes the Ferrari-Martin algorithm for TASEP, and we show it is equivalent to the algorithm of Arita-Ayyer-Mallick-Prolhac (2011). Our algorithm on bosonic multiline queues is novel and generalizes the corresponding algorithm of Kuniba-Maruyama-Okado (2016) for the TAZRP. We also define a Markov process on bosonic twisted multiline queues that projects to the TAZRP and intertwines with the symmetric group action on the rows of the multiline queues.