I believe the following exist:
- 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 $\mathbb{R}^d$.
- 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, but are abstractly isomorphic. The corresponding Lie groups are non-isomorphic as real Lie groups, because the Lie algebras are not isomorphic.