This is very similar to the answer of Mikhail Katz, but we can avoid the incompleteness theorem by using the halting problem instead.

That is, since the theory of real-closed fields is finitely axiomatizable and complete, it is decidable. So if $\mathbb{Z}$ were definable in $\langle\mathbb{R},+,\cdot,0,1,<\rangle$, then arithmetic truth would be decidable, contradicting the undecidability of the halting problem.

This argument still relies, however, on Tarski's quantifier-elimination.

This is very similar to the answer of Mikhail Katz, but we can avoid the incompleteness theorem by using the halting problem instead.

That is, since the theory of real-closed fields is computably axiomatizable and complete, it is decidable. So if $\mathbb{Z}$ were definable in $\langle\mathbb{R},+,\cdot,0,1,<\rangle$, then arithmetic truth would be decidable, contradicting the undecidability of the halting problem.

This argument still relies, however, on Tarski's quantifier-elimination.