Abstract: Context. Flat rotation curves, v(r), are naturally explained by elongated (prolate) dark matter (DM) distributions, and we have provided competitive fits to the SPARC database. To further probe the geometry of the halo, or the equivalent source of gravity in other formulations, one needs observables outside the galactic plane. Stellar streams, poetically analogous to airplane contrails, but caused by tidal dispersion of massive substructures such as satellite dwarf galaxies, would lie on their own plane (consistently with angular momentum conservation) should the DM-halo gravitational field be spherically symmetric. Tracks resembling entire orbits are seldom available because their periods are commensurable with Hubble time, with streams often presenting themselves as short segments. Aims. Therefore, we aim to establish stellar stream torsion, a local observable that measures the deviation from planarity in differential curve geometry, as a diagnostic providing sensitivity to aspherical DM distributions and ensuring the use of even relatively short streams. Methods. We performed small-scale simulations of tidally distorted star clusters to check that indeed a central force center produces negligible torsion, while distorted halos can generate it. Turning to observational data, we identified among the known streams those that are at the largest distance from the Galactic center, and that are likely not affected by the Magellanic clouds, as the most promising for the study, and by means of polynomial fits we extracted their differential torsion. Results. We find that the torsion of the few known streams that should be sensitive to most of the Milky Way's DM halo is much larger than expected for a central spherical bulb alone. This is consistent with the nonsphericity of the halo. Conclusions. Future studies of stellar stream torsion with larger samples and further out of the galactic plane should be able to extract the ellipticity of the halo to see whether it is just a slight distortion of a spherical shape or whether it rather resembles a more elongated cigar.