The usual Poincare-Hopf theorem for manifolds-with-boundary M expresses the Euler-Poincare characteristic of M as a sum of the local indices of the zeros of any vector field v exiting the boundary. Marston Morse (Amer. J. of Math. 51, 1929) and Charles Pugh (Topology 7, 1968) proved an extension allowing tangencies of v to the boundary. We generalise further to stratified-sets-with-boundary, and to even more general ‘radial manifold complexes’, including all finite partitions by definable submanifolds (in some o-minimal structure) inducing a filtration by closed subsets. The stratified vector fields need no longer be continuous. We introduce a notion of virtual zero and virtual index to treat generic vector fields, such as gradients of generic Morse functions. I will also discuss problems related to the stratifying of definable sets and maps.