I believe the following exist:

 1. A positive integer $d$ (anything $\geq 7$, I think), and a positive dimensional moduli space of unipotent groups of dimension $d$ defined over $\mathbb{Q}$.  All real points in this moduli space yield real Lie groups whose underlying manifolds are diffeomorphic to $\mathbb{R}^d$.
 2. A real quadratic extension $K/\mathbb{Q}$, and a pair of distinct $K$-points in this moduli space that are taken to each other by the nontrivial automorphism of $K$.

By fixing an embedding $K \to \mathbb{R}$, we obtain two real algebraic groups over these two $K$-points that are non-isomorphic as real algebraic groups.  The corresponding Lie groups are non-isomorphic as real Lie groups, because the Lie algebras are not isomorphic.  However, the nontrivial automorphism of $K$ induces an isomorphism on the underlying point-set groups.