  We use Borcherds products and their restrictions to Hirzebruch-Zagier curves
to determine generators and relations for the graded rings of Hilbert modular
forms for the fields $\mathbb{Q}(\sqrt{29})$ and $\mathbb{Q}(\sqrt{37})$. These
seem to be the first cases where the graded ring can be computed despite
obstructions to the existence of Borcherds products with arbitrary divisors.
