Coherent-injection Aronszajn tree ($T=\{e_{\alpha }\upharpoonright \beta :\beta \le \alpha <{\omega }_1\}$)

Given coherent coinfinite injections into omega, use all restrictions $e_{\alpha }\upharpoonright \beta$, where $\beta \le \alpha <{\omega }_1$, ordered by proper extension. Every level is countable because its nodes differ finitely from a fixed function on a countable domain. An uncountable chain in a partial order would have unbounded domain heights, and its union would inject ${\omega }_1$ into $\omega$, which is impossible.