Without further requirements on $\tau$, this is trivial. Let $T$ be any first-order theory and let $\top$ be any tautology. Define $\tau(\phi)= \top$.
If you want $\tau$ to be injective, then let $T$ be any first-order theory in the language of $TA$ and define $\tau(\phi)=\top\vee\phi$.
If you also want equivalence, then take $T$ to be any first-order theory in the language of $TA$ (such that $T\subseteq TA$) and define $\tau$ as follows:
-$\tau(\phi)=\phi$ if $\phi$ is not a theorem of $TA$
-$\tau(\phi)=\top\vee\phi$ if $\phi$ is a theorem of $TA$
This $\tau$ is injective and gives you the equivalence: $\forall \phi(TA\vdash\phi\leftrightarrow T\vdash\tau(\phi))$.
So, you must specify what do you mean by a suitable $\tau$, as Monroe Eskew has already asked in a comment.