Let \(G\) be a real reductive Lie group and \(G/H\) a reductive homogeneous space. The authors consider Kostants cubic Dirac operator \(D\) on \(G/H\) twisted with a finite-dimensional representation of \(H\). Under the assumption that \(G\) and \(H\) have the same complex rank, the authors construct a nonzero intertwining operator from principal series representations of \(G\) into the kernel of \(D\). The Langlands parameters of these principal series are described explicitly. In particular, they obtain an explicit integral formula for certain solutions of the cubic Dirac equation \(D=0\) on \(G/H\).