The origin of high-energy cosmic neutrinos detected by the IceCube observatory is a hotly debated topic in astroparticle physics. There is growing evidence that some of these neutrinos can be associated with active galactic nuclei (AGN) and especially with blazars. Several recent studies have revealed a statistical correlation between radio-bright AGN samples and IceCube neutrino event catalogs. In addition, a growing number of individual high-energy neutrinos have been found to coincide with individual radio-flaring blazars. These observational results strongly call for high-quality, high angular-resolution radio observations of such neutrino-associated blazars to study their parsec-scale jet structures. TANAMI is the only large and long-term VLBI monitoring program focused on the Southern sky. Within TANAMI, we put an emphasis on Southern IceCube neutrino candidate blazars at 2.3 GHz and 8.4 GHz. Here we present first results of the first high-quality, high angular-resolution VLBI observations of nine Southern-Hemisphere blazars that were associated to IceCube neutrino hotspots in the Southern sky. In the near future, the rapidly growing KM3NeT will complement IceCube by being sensitive to high-energy neutrinos mainly from the Southern Hemisphere. This will increase the importance of Southern-Hemisphere radio monitoring programs of neutrino-associated blazars, like TANAMI.