Spin structures and phase diagrams of the spin- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mn>5</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:math> triangular lattice antiferromagnet <mml:math xmlns:mml="http://www.w3

We combine single-crystal neutron diffraction studies and Monte Carlo simulations to determine the spin structures and finite-temperature phase diagram of the spin-5/2 triangular-lattice antiferromagnet ${\mathrm{Na}}_{2}\mathrm{BaMn}{({\mathrm{PO}}_{4})}_{2}$ in magnetic field. With the application of a magnetic field in two different directions, namely along the $c$ axis and in the $ab$ plane of the trigonal symmetry, we track the evolution of the spin structure through changes of the magnetic propagation vector. We account for these results with a minimal Heisenberg Hamiltonian that includes easy-axis anisotropy and weak, frustrated interlayer couplings in addition to intralayer exchange. Guided by representation analysis, we refine symmetry-allowed modes to the measured intensities and obtain the spin structures for all field-induced phases, which we compare quantitatively with simulated configurations. Taken together, our measurements and simulations show that frustrated interlayer exchange---rather than purely two-dimensional physics---organizes the unexpectedly rich field-induced phases of ${\mathrm{Na}}_{2}\mathrm{BaMn}{({\mathrm{PO}}_{4})}_{2}$.