Due to its proximity, SN 1987A offers a unique opportunity to directly observe the geometry of a stellar explosion as it unfolds. Here we present spectral and imaging observations of SN 1987A obtained ˜10,000 days after the explosion with HST/STIS and VLT/SINFONI at optical and near-infrared wavelengths. These observations allow us to produce the most detailed 3D map of Hα to date, the first 3D maps for [Ca ii] λ λ 7292,7324, [O i] λ λ 6300,6364, and Mg ii λ λ 9218,9244, as well as new maps for [Si i]+[Fe ii] 1.644 μ {{m}} and He i 2.058 μm. A comparison with previous observations shows that the [Si i]+[Fe ii] flux and morphology have not changed significantly during the past ten years, providing evidence that this line is powered by ⁴⁴Ti. The time evolution of Hα shows that it is predominantly powered by X-rays from the ring, in agreement with previous findings. All lines that have sufficient signal show a similar large-scale 3D structure, with a north-south asymmetry that resembles a broken dipole. This structure correlates with early observations of asymmetries, showing that there is a global asymmetry that extends from the inner core to the outer envelope. On smaller scales, the two brightest lines, Hα and [Si i]+[Fe ii] 1.644 μm, show substructures at the level of ˜200-1000 km s^-1and clear differences in their 3D geometries. We discuss these results in the context of explosion models and the properties of dust in the ejecta.