  We show that the dynamics of sufficiently dissipative semi-Siegel complex
H\'enon maps with golden-mean rotation number is not $J$-stable in a very
strong sense. By the work of Dujardin and Lyubich, this implies that the
Newhouse phenomenon occurs for a dense $G_\delta$ set of parameters in this
family. Another consequence is that the Julia sets of such maps are
disconnected for a dense set of parameters.
