Activated muscle applies forces in response to its stimulus but also when mechanically perturbed. The temporal characteristics of decay in stresses due to step perturbations are key to characterizing the mechanical response. If stresses are long-lived, muscle is stiff like an elastic solid and could be used in posture maintenance tasks. But if stresses are rapidly dissipated, muscle yields like a viscous liquid and could be used for rapidly changing posture. We show by modeling muscles as half-sarcomeres with cycling crossbridges on a compliant filament backbone that filament compliance leads to an increase in the stress relaxation time compared with a rigid backbone. The slowdown depends on a single dimensionless ratio l/p of the filament overlap l to a length scale p that arises from a partitioning of strain between the filament backbone and crossbridges. In living muscle, this parameter would be varied by the mean number of attached crossbridges so that greater activation implies a greater value of (l/p). The relaxation time constant is nearly constant for l/p << 1, but increases nonlinearly as (l/p)^2 when (l/p)>>1. We estimate (l/p)>5 for mammalian sarcomeres with a 10x slowdown of stress relaxation. The physical basis lies in interactions between crossbridges. With a compliant filament, the attachment of one crossbridge distorts neighboring crossbridges and induces changes in its kinetics. Many muscle models thus far have assumed a rigid filament backbone, which prevents this mode of cooperativity. Thus, the effects introduced by filament compliance may be a possible mechanism underlying a muscle’s ability to maintain an elastic behavior on timescales much longer than crossbridge cycling. These effects may scale up to the whole fiber based on titin that mediates inter-sarcomeric interactions.