Models of VTC 0 $\mathsf {VTC^0}$ as exponential integer parts

We prove that (additive) ordered group reducts of nonstandard models of the bounded arithmetical theory ${\mathrm{VTC}}^0$ 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 ${\mathrm{VTC}}^0$, we show that every countable model of ${\mathrm{VTC}}^0$ is an exponential integer part of a real-closed exponential field.