  We prove that (additive) ordered group reducts of nonstandard models of the
bounded arithmetical theory $\mathsf{VTC^0}$ are recursively saturated in a
rich language with predicates expressing the integers, rationals, and
logarithmically bounded numbers. Combined with our previous results on the
construction of the real exponential function on completions of models of
$\mathsf{VTC^0}$, we show that every countable model of $\mathsf{VTC^0}$ is an
exponential integer part of a real-closed exponential field.
