We describe a new algorithm for the perfect simulation of variable length Markov chains and random systems with perfect connections. This algorithm generalizes Propp and Wilson's simulation scheme, and is based on the idea of coupling into and from the past. It improves on existing algorithms by relaxing the conditions required on the kernel and by accelerating convergence, even in the simple case of finite order Markov chains. Chains of variable or infinite order are an old object of consideration that raised considerable interest recently because of their applications in applied probability, from information theory to bio-informatics and linguistics.