[Goncharov, Harizanov, Knight and Shore][1] investigated the Turing degrees of $\Pi^1_1$ cofinal branches (which they call "paths through $\mathcal{O}$").  They showed there is a $\Pi^1_1$ cofinal branch which does not compute $\emptyset'$, so certainly doesn't compute true arithmetic.  On the other hand, there is a $\Pi^1_1$ cofinal branch which computes $\mathcal{O}$.


  [1]: http://pi.math.cornell.edu/~shore/papers/pdf/GHKSFinalVersion3.pdf