It is well-known that $\mathbf{PRA}$ plus $\epsilon_0$-induction on bounded formulas cannot prove all $\mathbf{PA}$ theorems (essentially because $I\Sigma_1$ plus $\epsilon_0$-induction on bounded formulas is finitely axiomatizable while the latter isn't). Are there any concrete examples known (preferable "natural")?

(This is an exact copy of my question on MSE, but I expect I won't get any answer there, so I hope I'm allowed to cross-post it here)