  Consider a homogeneous multifold convex conic system $$ Ax = 0, \; x\in
K_1\times...\times K_r $$ and its alternative system $$ A\transp y \in
K_1^*\times...\times K_r^*, $$ where $K_1,..., K_r$ are regular closed convex
cones. We show that there is canonical partition of the index set ${1,...,r}$
determined by certain complementarity sets associated to the most interior
solutions to the two systems. Our results are inspired by and extend the
Goldman-Tucker Theorem for linear programming.
