The application of flying systems to practical tasks is consistently limited by the poor endurance of hovering robots. The ability to perch to fixed surfaces allows a robot to gather data and inspect structures in a low power state, while retaining the access and manoeuvrability that flight offers. In this paper we present a passively adaptive perching mechanism which allows an aerial vehicle to stably attach to a variety of surfaces including tree branches and pipelines. This is enabled by a compliant grapple module, which passively conforms to the surface of convex perching targets, ensuring reliable traction and a very high load capacity (tension tested to > 60 kg in some instances) whilst still releasing effortlessly. This is due to the mechanics of the grapple, which is designed to passively tighten and attach to a variety of branch diameters and shapes. The grapple is paired with a hybrid force-motion controller which allows the cable tension to be regulated as the vehicle achieves the desired attitude. The hybrid control approach exploits the mechanical compliance of the system to ensure reliable, stable attachment to irregular natural structures, and the addition of a winch allows the robot to stably orient itself in any position or orientation relative to the branch. This approach demonstrates tensile perching using adaptive anchors. The presented subsystems can be applied to other robots where high force authority is required.