Real lines tangent to 2n-2 quadrics in Rⁿ

Let C_n,k be the number of k-flats in Pⁿ which meet N := (k+1)(n-k) general linear subspaces of codimension k. We show that for each n>k there are 2^N C_n,k k-flats in Pⁿ tangent to N general quadric hypersurfaces in Pⁿ, and that there exists a configuration of N real quadrics so that all the mutually tangent k-flats are real.