This paper develops what is essentially a simplified version of the block-iterative operator splitting method already proposed by the author and P. Combettes, but with more general initialization conditions. It then describes one way of implementing this algorithm asynchronously under a computing model inspired by modern HPC environments, which consist of interconnected nodes each having multiple processor cores sharing a common local memory. The asynchronous implementation framework is then applied to derive an asynchronous algorithm which resembles the ADMM with an arbitrary number of blocks of variables. Unlike earlier proposals for asynchronous ADMM-like methods, the algorithm relies neither on probabilistic control nor on restrictive assumptions about the problem instance, instead making only standard convex-analytic regularity assumptions. It also allows the proximal parameter to vary freely between arbitrary positive bounds, an unusual feature in an ADMM-like method.