We investigate the vectorial nature of soliton fission in an isotropic nonlinear medium both theoretically and experimentally. As a specific example, we show that supercontinuum generation in a tapered fiber is extremely sensitive to the input state of polarization. Multiple vector solitons generated through soliton fission exhibit different states of elliptical polarization while emitting nonsolitonic radiation with complicated polarization features. Experiments performed with a tapered fiber agree with our theoretical description.