The answer to the question is in the positive (provided "real ordered field" is interpreted as "real closed field").
The details are worked out in a section 3 of a paper by Salehi and Zarza (a preprint of which can be found here); the paper was published as follows:
S. Salehi & M. Zarza, First-Order Continuous Induction and a Logical Study of Real Closed Fields, Bulletin of the Iranian Mathematical Society volume 46, pages225–243(2020).