Real froms of complex unipotent groups

Question

Let $G$ be a real linear group such that $G^{\mathbb C}$ is its complexification group in $GL_n(\mathbb C)$. Let $U$ be a unipotent complex Lie subgroup of $G^{\mathbb C}$. Does $U$ have a real form i.e. $(U\cap G)^{\mathbb C}=U$?