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Developing efficient MCMC algorithms is indispensable in Bayesian inference. In parallel tempering, multiple interacting MCMC chains run to more efficiently explore the state space and improve performance. The multiple chains advance independently through local moves, and the performance enhancement steps are exchange moves, where the chains pause to exchange their current sample amongst each other. To accelerate the independent local moves, they may be performed simultaneously on multiple processors. Another problem is then encountered: depending on the MCMC implementation and inference problem, local moves can take a varying and random amount of time to complete. There may also be infrastructure-induced variations, such as competing jobs on the same processors, which arises in cloud computing. Before exchanges can occur, all chains must complete the local moves they are engaged in to avoid introducing a potentially substantial bias (Proposition 2.1). To solve this issue of randomly varying local move completion times in multi-processor parallel tempering, we adopt the Anytime Monte Carlo framework of Murray et al. (2016): we impose real-time deadlines on the parallel local moves and perform exchanges at these deadlines without any processor idling. We show our methodology for exchanges at real-time deadlines does not introduce a bias and leads to significant performance enhancements over the naïve approach of idling until every processor's local moves complete. The methodology is then applied in an ABC setting, where an Anytime ABC parallel tempering algorithm is derived for the difficult task of estimating the parameters of a Lotka-Volterra predator-prey model, and similar efficiency enhancements are observed.