Spin textures in k-space arising from spin-orbit coupling in noncentrosymmetric crystals find numerous applications in spintronics. We present a mechanism that leads to the appearance of k-space spin texture due to spontaneous symmetry breaking driven by electronic correlations. Using dynamical mean-field theory we show that doping a spin-triplet excitonic insulator provides a means of creating new thermodynamic phases with unique properties. The numerical results are interpreted using analytic calculations within a generalized double-exchange framework.