Actin filaments associated with myosin motors constitute the cytoskeletal force-generating machinery for many types of adherent cells. These actomyosin units are structurally ordered in muscle cells and, in particular, may be spatially registered across neighboring actin bundles. Such registry or stacking of myosin filaments have been recently observed in ordered actin bundles of even fibroblasts with super-resolution microscopy techniques. We introduce here a model for the dynamics of stacking arising from long-range mechanical interactions between actomyosin units through mutual contractile deformations of the intervening cytoskeletal network. The dynamics of registry involve two key processes: 1) polymerization and depolymerization of actin filaments and 2) remodeling of cross-linker-rich actin adhesion zones, both of which are, in principle, mechanosensitive. By calculating the elastic forces that drive registry and their effect on actin polymerization rates, we estimate a characteristic timescale of tens of minutes for registry to be established, in agreement with experimentally observed timescales for individual kinetic processes involved in myosin stack formation, which we track and quantify. This model elucidates the role of actin turnover dynamics in myosin stacking and explains the loss of stacks seen when actin assembly or disassembly and cross-linking is experimentally disrupted in fibroblasts.