Countryman basis and club-isomorphisms under MA$({\omega }_1)$

We show that unlike $\mathsf{\text{PFA}}$, ${\mathsf{\text{MA}}}_{{\omega }_1}$ is consistent with the existence of $2^{{\omega }_1}$-many Countryman lines which pairwise contain no uncountable isomorphic or reverse-isomorphic suborders. This clarifies an issue regarding the claim that ${\mathsf{\text{MA}}}_{{\omega }_1}$ implies that there is a finite basis for the Countryman lines. The family of Countryman lines comes from a family of $2^{{\omega }_1}$-many coherent trees which pairwise contain no club-isomorphic subtrees. To prove this, we first construct such a family of trees using $\mathrm{♢}$, and then we iterate ccc forcings while preserving this property.