Consider a dissipative quadratic Hénon map $H_{\mu, \nu}$ that has a fixed point in $\mathbb{C}^2$ with multipliers $\mu$ and $\nu$. As $\nu$ goes to $0$, the dynamics of $H_{\mu, \nu}$ degenerates to that of the quadratic polynomial $f_\mu$ that has a fixed point in $\mathbb{C}$ with multiplier $\mu$. Due to work of Hubbard and Oberste-Vorth, and Radu and Tanase respectively, the structure of the Julia set $J^+$ of $H_{\mu, \nu}$ was shown to be stable in the hyperbolic ($|\mu| < 1$) and the semi-parabolic case ($\mu$ is a rational rotation). Using a renormalization approach inspired by the work of Lyubich and Martens for Feigenbaum Hénon maps, we show that structural stability fails in the golden-mean semi-Siegel case ($\mu$ is the golden-mean rotation). This is joint work with M. Yampolsky.