Circular tubes compressed into the plastic range first buckle into axisymmetric wrinkling. Initially, the wrinkle amplitude grows with increasing load, but induces a gradual reduction in axial rigidity that eventually leads to a limit load instability and collapse. For lower $D/t$ tubes, the two instabilities can be separated by strain levels of a few percent. Persistent stress-controlled cycling can cause accumulation of deformation by ratcheting. Here, the interaction of ratcheting and wrinkling is investigated. In particular, it is asked if compressive ratcheting can first initiate wrinkling and then grow it to amplitudes associated with collapse. Experiments on SAF2507 super-duplex steel tubes with $D/t$ of 28.5 have shown that a geometrically intact tube cycled under stress control initially deforms uniformly due to material ratcheting. However, in the neighborhood of the critical wrinkling strain under monotonic loading, small amplitude axisymmetric wrinkles develop. This happens despite the fact that the maximum stress of the cycles can be smaller than the critical stress under monotonic loading. In other words, wrinkling appears to be strain rather than stress driven, as is conventionally understood. Once the wrinkles are formed, their amplitude grows with continued cycling, and as a critical value of amplitude is approached, wrinkling localizes, the rate of ratcheting grows exponentially, and the tube collapses. Interestingly, collapse was also found to occur when the accumulated average strain reaches the value at which the tube localizes under monotonic compression. A custom shell model with small initial axisymmetric imperfections, coupled to a cyclic plasticity model, is used to simulate these cyclic phenomena successfully.