Using the largest three-dimensional particle-in-cell simulations to date, collisionless magnetic reconnection in large-scale electron-positron plasmas without a guide field is shown to involve complex interaction of tearing and kink modes. The reconnection onset is patchy and occurs at multiple sites which self-organize to form a single, large diffusion region. The diffusion region tends to elongate in the outflow direction and become unstable to secondary kinking and formation of "plasmoid-rope" structures with finite extent in the current direction. The secondary kink folds the reconnection current layer, while plasmoid ropes at times follow the folding of the current layer. The interplay between these secondary instabilities plays a key role in controlling the time-dependent reconnection rate in large-scale systems.