My original conception of Reverse Mathematics was actually of Strict Reverse Mathematics. SRM can be viewed either as the requirement that there be no base theory at all, or the requirement that every statement of the base theory be itself strictly mathematical in its own right. Although these two formulations can be viewed as equivalent, the second formulation suggests a lot of systematic reuse of some fundamental strictly mathematical statements. This is expected.
Already in the founding JSL abstracts of RM, there was a claim in passing that RCA_0 is logically equivalent to a strictly mathematical theory called ETF (elementary theory of functions), without proof. Since RM based on RCA_0 was so readily accepted I left that claim undocumented. A proof appears in https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/ #116. In #116 there are also some other results in SRM which I will discuss.
Forging SRM into a systematically vibrant subject like RM is a challenge that needs to be met. This is much more difficult than for RM. There are some moves in this direction in #116 which we are now in the process of upgrading and extending. I will be reporting on the state of the art.