The temperature-magnetic-field phase diagram of the mixed honeycomb triangular lattice system K$_{2}$Mn$_{3}$(VO$_{4}$)$_{2}$CO$_{3}$ is investigated by means of magnetization, heat capacity and neutron scattering measurements. The results indicate that triangular and honeycomb magnetic layers undergo sequential magnetic orderings and act as nearly independent magnetic sublattices. The honeycomb sublattice orders at about 85 K in a Ne{e}l-type antiferromagnetic structure, while the triangular sublattice displays two consecutive ordered states at much lower temperatures, 3 K and 2.2 K. The ground state of the triangular sublattice consists of a planar `Y magnetic structure that emerges from an intermediate collinear `up-up-down state. Applied magnetic fields parallel or perpendicular to the $c$-axis induce exotic ordered phases characterized by various spin-stacking sequences of triangular layers that yield bilayer, three-layer or four-layer magnetic superstructures. The observed superstructures cannot be explained in the framework of quasi-classical theory based only on nearest-neighbor interlayer coupling and point towards the presence of effective second-nearest-neighbor interactions mediated by fluctuations of the magnetic moments in the honeycomb sublattice.
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