  The grassmannian of hermitian lagrangian spaces in $\mathbb{C}^n\oplus
\mathbb{C}^n$ is a natural compactification of the space of hermitian $n\times
n$ matrices. We describe a Schubert-like, Whitney regular stratification on
this space which has a Morse theoretic origin. We prove that these strata
define closed subanalytic currents \`{a} la R. Hardt, generating the integral
homology of this space, we investigate their intersection theoretic properties,
and we prove certain odd (in K-theoretic sense) Thom-Porteous type theorems.
