  In this paper, we investigate the rigidity of the stable comodule category of
a specific class of Hopf algebroids known as finite Adams, shedding light on
its Picard group. Then we establish a reduction process through base changes,
enabling us to effectively compute the Picard group of the $\mathbb{R}$-motivic
mod $2$ Steenrod subalgebra $\mathcal{A}(1)^{\mathbb{R}}$. Our computation
shows that $\operatorname{Pic}(\mathcal{A}(1)^{\mathbb{R}})$ is isomorphic to
$\mathbb{Z}^4$, where two ranks come from the motivic grading, one from the
algebraic loop functor, and the last is generated by the $\mathbb{R}$-motivic
joker $J$.
