Explicit Hecke eigenform product identities for Hilbert modular forms

Let $F$ be a totally real number field, and $g,f,h$ be Hilbert modular forms over $F$ that are Hecke eigenforms satisfying $g=f\cdot h$. We characterize such product identities among all real quadratic fields of narrow class number one, proving they occur only for $F=\mathbb Q(\sqrt{5})$, with precisely two such identities. We also shed some light on the general totally real case by showing that no such identity exists when both $f$ and $h$ are Eisenstein series of distinct weights.