While thinking about the sound changes leading up to Ćwarmin and Ŋʒädär, an uninvited idea entered my mind and turned into this post: what can we do to combine differential subject/object marking and inverse alignment?
The idea might seem somewhat preposterous. Differential subject marking obviously acts on the subject, differential object marking on the object, and inverse alignment doesn't formally distinguish the two in any case-like way.
However, maybe there are ways out, and maybe these can have interesting restrictions on them.
Since I have not mentioned differential case in quite a while, differential case is where subjects (or objects) can take different cases depending on some syntactical or semantic factors.
A few options:
1. mutual marking
Both nouns take the same marker (or allomorphs thereof). This could fit for some Finnish-style differential object marking, i.e. conveying aspectual and/or polarity-related information, but is not suitable for Turkish-style differential marking (where the accusative marks the definiteness of a direct object, whereas the nominative implies indefiniteness).
Jazzing it up: cancel the use of the inverse (or maybe the direct) whenever the Differential Subject&Object Marking is in place. Only let a certain assignment of roles to the two nouns be permissible whenever the marking is there. Here, we can get verb-specific things going: maybe one verb forces inverseness whenever differential marking is in place, maybe another forces directness.
2. marking determined by the hierarchy
One could imagine that the marker goes on whichever noun is lower (or higher) on the hierarchy. Maybe several markers can coexist, and two different ones are permissible, as long as they do not try to go on the same noun. Maybe some markers can go on either noun, and some are restricted to the higher or the lower one. An additional complication could be markers that only exist on some level of the hierarchy - i.e. markers only present with first and second person pronouns, or markers only present for inanimate nouns.
Jazzing it up: have some markers go on 'actual subject' or 'actual object' and some markers follow the above rule. Have a hierarchy among the markers as to which beats which.
3. marking determined by topicality or focus
Whichever element is topic, or potentially focus, could be differentially marked, while the other one goes unmarked.
Jazzing it up: again, change how the verb interacts with things. Make the verb intransitive, and the 'subject' an oblique agent or the object and oblique patient, depending on which one is the topic. This would functionally make the intransitive be both a passive and an antipassive at once. Heck, for double topics, you could have a simultaneous passive-antipassive interpretation.