Recent field experiments conducted in the near wake (up to 0.5 rotor diameters downwind of the rotor) of a Clipper Liberty C96 2.5 MW wind turbine using snow-based super-large-scale particle image velocimetry (SLPIV) (Hong et al., Nat. Commun., vol. 5, 2014, 4216) were successful in visualizing tip vortex cores as areas devoid of snowflakes. The so-visualized snow voids, however, suggested tip vortex cores of complex shape consisting of circular cores with distinct elongated comet-like tails. We employ large-eddy simulation (LES) to elucidate the structure and dynamics of the complex tip vortices identified experimentally. We show that the LES, with inflow conditions representing as closely as possible the state of the flow approaching the turbine when the SLPIV experiments were carried out, reproduce vortex cores in good qualitative agreement with the SLPIV results, essentially capturing all vortex core patterns observed in the field in the tip shear layer. The computed results show that the visualized vortex patterns are formed by the tip vortices and a second set of counter-rotating spiral vortices intertwined with the tip vortices. To probe the dependence of these newly uncovered coherent flow structures on turbine design, size and approach flow conditions, we carry out LES for three additional turbines: (i) the Scaled Wind Farm Technology (SWiFT) turbine developed by Sandia National Laboratories in Lubbock, TX, USA; (ii) the wind turbine developed for the European collaborative MEXICO (Model Experiments in Controlled Conditions) project; and (iii) the model turbine presented in the paper by Lignarolo et al. (J. Fluid Mech., vol. 781, 2015, pp. 467–493), and the Clipper turbine under varying inflow turbulence conditions. We show that similar counter-rotating vortex structures as those observed for the Clipper turbine are also observed for the SWiFT, MEXICO and model wind turbines. However, the strength of the counter-rotating vortices relative to that of the tip vortices from the model turbine is significantly weaker. We also show that incoming flows with low level turbulence attenuate the elongation of the tip and counter-rotating vortices. Sufficiently high turbulence levels in the incoming flow, on the other hand, tend to break up the coherence of spiral vortices in the near wake. To elucidate the physical mechanism that gives rise to such rich coherent dynamics we examine the stability of the turbine tip shear layer using the theory proposed by Leibovich & Stewartson (J. Fluid Mech., vol. 126, 1983, pp. 335–356). We show that for all simulated cases the theory consistently indicates the flow to be unstable exactly in the region where counter-rotating spirals emerge. We thus postulate that centrifugal instability of the rotating turbine tip shear layer is a possible mechanism for explaining the phenomena we have uncovered herein.