  We study the Iwasawa main conjecture for quadratic Hilbert modular forms over
the p-cyclotomic tower. Using an Euler system in the cohomology of Siegel
modular varieties, we prove the "Kato divisibility" of the Iwasawa main
conjecture under certain technical hypotheses. By comparing this result with
the opposite divisibility due to Wan, we obtain the full Main Conjecture over
the cyclotomic Zp-extension.
  As a consequence, we prove new cases of the Bloch--Kato conjecture for
quadratic Hilbert modular forms, and of the equivariant Birch--Swinnerton-Dyer
conjecture in analytic rank 0 for elliptic curves over real quadratic fields
twisted by Dirichlet characters.
