Let $T$ be a r.e. and consistent extension of **PA** in the language of artihmetic. Are there examples of arithmetical sentences $\phi, \psi$ such that $T+\phi \equiv\_{\Pi^0\_1} T+ \psi$ but $T \not \vdash Con\_{\tau + \phi} \leftrightarrow Con\_{\tau + \psi}$, where $\tau$ numerates the $T$-axioms in $T$ and $\tau + \phi := \tau(x) \vee x = \overline{\phi} $ (respect. $\psi$)?



Unprovable cases of $\Pi^0_1$-conservativity
I'd say that this situation is indeed possible. In general, the $\Pi\_1^0$-conservativity of one theory over another does not guarantee that a proof of their relative consistency can be formalised in $PA$. This observation is due to Guaspari ([Partially conservative extensions of arithmetic](https://doi.org/10.1090/S0002-9947-1979-0539907-7), Transactions of the AMS, Vol. 254, 1979, pp. 47-68, Theorem 3.3). Recall that $\Pi\_1^0$-conservativity and interpretability coincide for essentially reflexive, r.e. theories.


**Theorem** (Guaspari). Let $T$ be r.e. and essentially reflexive. Then there is a $\Sigma\_1^0$ sentence $\phi$ such that $T + \phi$ is interpretable in $T$, but $PA \nvdash Con(T) \rightarrow Con(T+\phi)$.


