  We continue our investigation on cluster algebras arising from cluster tubes.
Let $\mathcal{C}$ be a cluster tube of rank $n+1$. For an arbitrary basic
maximal rigid object $T$ of $\mathcal{C}$, one may associate a
skew-symmetrizable integer matrix $B_T$ and hence a cluster algebra
$\mathcal{A}(B_T)$ to $T$. We define an analogue Caldero-Chapoton map
$\mathbb{X}_M^T$ for each indecomposable rigid object $M\in \mathcal{C}$ and
prove that $\mathbb{X}_?^T$ yields a bijection between the indecomposable rigid
objects of $\mathcal{C}$ and the cluster variables of the cluster algebra
$\mathcal{A}(B_T)$. The construction of the Caldero-Chapoton map involves
Grassmanians of locally free submodules over the endomorphism algebra of $T$.
We also show that there is a non-trivial $\mathbb{C}^{\times}$-action on the
Grassmanians of locally free submodules, which is of independent interest.
