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Yes. As an Abelian group, the reals are a nontrivial torsion-free divisible group, equivalently a vector space over the rationals. Rational scalar multiplication is uniquely definable by equations such as $ny=mx$, so adding those operations is a definitional expansion.

In this language, quantifier elimination for torsion-free divisible Abelian groups reduces formulas to Boolean combinations of rational linear equations. Over finitely many parameters, a complete one-variable type either fixes a value in their linear span or excludes every value in that span. The first kind is realized in the span itself; the second is realized by any real number outside it. That span is countable, while the reals are uncountable. Thus every such type is realized and the additive real group is an omega-saturated model.

For finite tuples, the same argument successively supplies any required finite number of new rationally linearly independent elements. The order and multiplication of the real field are absent from this language, so they cannot impose additional type conditions.