Let us consider a language that, like Finnish or Russian, marks the agent-like argument of 'to have to' with a non-canonical case. It is arguable whether such a noun is a subject, and the claim is that in Russian at least they aren't, but they pass subjecthood tests in Finnish rather splendidly.
Now, both the Russian and Finnish constructions lack subject marking entirely on the VP. and in some sense maybe we could say this is a result of the nominative controlling congruence on verbs in these languages.
Now, let's keep that situation intact with regards to finite verbs, but introduce a conflicting notion for certain infinitives! In Finnish, there are several different constructions for obligation, some using specific verbs, and one using the 3rd person sg. copula in combination with the passive present participle. Thus, mine is eat-en-ing sort of expresses 'I have to eat'.
Some languages have person marking on infinitives - examples include Portuguese. Now, we can imagine the semantics of the situation forcing an explicit subject marking on such infinitives even when the subject is not nominative (or maybe even when the agent is not a subject), even when the finite verb does have no subject marking.
Now we can start imagining somewhat weirder stuff. Where do these subject markers originate? Possibly with possessive markers (and further down the line with pronouns), right? Some languages have reflexive possessives distinct from regular possessives - Scandinavian sin, Slavic swój, etc. We can imagine such pronouns also to become part of a system of possessive affixes, thus not only giving us 'mine, yours, his/hers/its, ours, yours, theirs" as affixes, but also a distinct 'his/hers/its own' as a suffix.
In Scandinavian, the reflexive possessive is only used with third person possessors, so
Stina sålde sitt hus
Stina sold her (own, not someone else's) house
consider, in English, conversely
Etta was angry after Stina had sold her house
Elna var arg efter att Stina sålde hennes hus
Thus, this lets us distinguish two potential possessors in Scandinavian, but English requires at least one extra word, and the presence of that extra word ('own') is sometimes ambiguous even at that. And even omitting 'own' does not exclude a reflexive parsing, and getting the non-reflexive parsing explicit can be awkward.
Anyways, in Slavic languages, swój and its cognates are used for all persons. So, if 'I sell my portfolio', it is я who sells свой portfolio.
Back to the infinitives: now we can imagine a situation where a quirky case construction with an infinitive verb has a reflexive subject. We can, for instance, consider a situation where 'has to' and 'needs' are the same verb, and an embedded participle, for instance, can take a different subject:
me-obl [has to, needs, must] hear-2sg this-acc
I need you to hear this
she-obl [needs, ...] hearing-refl this-acc
she needs to hear this
she-obl [needs, ...] hearing-3sg this-acc
she1 needs for him/her2 to hear this
Another thing this could introduce is differential subject marking! First and second person embedded infinitives could go either way with regards to having 'canonical' pronominal agreement or reflexive agreement, and communicate, say, volition or somesuch in that way.