In Jech's Set theory (2003) refer to the Aronszajn tree that has real-valued increasing function. Since $\mathsf{ZFC+MA}$ proves that every Aronszajn tree is special, so 'every Aronszajn tree has real-valued increasing function' is consistent with $\mathsf{ZFC}$.
Question. $\mathsf{ZFC}$ can prove the existence of increasing real-valued function on every Aronszajn tree? that is, if $T$ is Aronszajn tree then there is a function $f:T\to\Bbb R$ such that $f(x)<f(y)$ whenever $x<y$?
Thanks for any help.
It is a theorem of Devlin (Note on a theorem of J. Baumgartner, Fund. Math. 76, 1972) that for any $\mathbb{R}$-embeddable $\omega_1$-tree $T$, any uncountable subset $U$ of $T$ must contain an uncountable antichain. This follows since the union of the successor levels of $U$ is in fact $\mathbb{Q}$-embeddable and thus the union of countably many antichains. So Asaf's suggestion in the comments about Suslin trees being a counterexample is correct.
In the same paper he also constructs an $\mathbb{R}$-embeddable nonspecial Aronszajn tree from $\diamondsuit$. Furthermore, there is a slightly earlier result of Baumgartner where he get a non-Suslin non-$\mathbb{R}$-embeddable Aronszajn tree in $L$.