= Solution
\b[No.] The parameter-free <partial type>
$$
\{x>n\cdot1:n\in\mathbb N\}
$$
is finitely satisfiable in the ordered real field: choose a number larger than the finitely many bounds. By <compactness theorem> it extends to a <complete type> over the empty set. But no real number realizes it, by the <Archimedean property>. Therefore the ordered real field is not an <omega-saturated model>.
The named unit and addition make every integer bound expressible, even without extra parameters. This is why the obstruction is present in the ordered-field language and absent in the two reducts above.
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