Answer by Arno for Is the union of two conservative extensions of a theory conservative?

The answer is no. Let $\phi$ be a non-theorem in the language of $T$, and let $c,d$ be two fresh symbols for constants. Now we can take $T_1$ to add the axiom $(c = d) \Rightarrow \phi$, and $T_2$ to add the axiom $c = d$.