(This is a follow-up to this question of mine.)

Is there an example of a connected reductive algebraic group $G$ over $\mathbb{R}$ such that:

• $G$ is not isomorphic to a product $G_1 \times G_2$ of smaller groups (isogenous to a product is OK)
• $G$ is not a torus,
• $Z_G(\mathbb{R})$ is not contained in the identity component of $G(\mathbb{R})$?

The condition $Z_G(\mathbb{R}) \subseteq G(\mathbb{R})^\circ$ is vacuously satisfied if $G$ is adjoint, because then $Z_G = \{1\}$; but it is also vacuously satisfied if $G$ is semisimple and simply-connected, because then $G(\mathbb{R})$ is connected as a Lie group by a theorem of Cartan. So any example would have to lie somewhere in between the two (which makes me wonder if there are any examples at all).

PS: Of course $GL_3$ is an example (oops). However, I'm particularly interested in cases where the symmetric space for $G$ has a complex structure.