Active cytoskeletal materials in vitro demonstrate self-organizing properties similar to those observed in their counterparts in cells. However, the search to emulate phenomena observed in living matter has fallen short of producing a cytoskeletal network that would be structurally stable yet possess adaptive plasticity. Here, we address this challenge by combining cytoskeletal polymers in a composite where self-assembling microtubules and actin filaments collectively self-organize due to the activity of microtubule-percolating molecular motors. We demonstrate that microtubules spatially organize actin filaments that in turn guide microtubules. The two networks align in an ordered fashion using this feedback loop. In this composite, actin filaments can act as structural memory and, depending on the concentration of the components, microtubules either write this memory or get guided by it. The system is sensitive to external stimuli, suggesting possible autoregulatory behavior in changing mechanochemical environments. We thus establish an artificial active actin–microtubule composite as a system demonstrating architectural stability and plasticity.