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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 basicmaximal rigid object $T$ of $\mathcal{C}$, one may associate askew-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 $M\in \mathcal{C}$ andprove that $\mathbb{X}_?^T$ yields a bijection between the indecomposable rigidobjects of $\mathcal{C}$ and the cluster variables of the cluster algebra$\mathcal{A}(B_T)$. The construction of the Caldero-Chapoton map involvesGrassmanians of locally free submodules over the endomorphism algebra of $T$.We also show that there is a non-trivial $\mathbb{C}^{\times}$-action on theGrassmanians of locally free submodules, which is of independent interest.

Cluster algebras arising from cluster tubes II: the Caldero-Chapoton map