I have (a) carHowever, to spice this up each noun has an associated preposition*. In other TAM/etc configurations, this prepositions surfaces. This serves to increase the amount of redundancy in the language, and is basically a congruence marker - and since it codes for two things [1) a type of noun class, 2) verb TAM] it might be rather an effective strategy with regards to increasing congruence. Now, there's several places this preposition could surface. The first, and obvious one, is with the possessed noun.
I had with a carOf course, we could have case:preposition:etc congruence behave oddly for these if we wanted to distinguish them from situations where we for some reason want to say that in collusion with a car, we had another object. Word order could of course also help figure out which noun is the possessum in case several prepositional phrases occur if we want that to be a strategy here.
John had to an idea
You had by an apartment in the town, right?
Next up is the idea of the preposition being independent of any NPs - either like a German separable prefix or like some kind of general adverbial. Not all that fascinating, but you can probably see it happening:
Erin had a boat byPrefixing it to the verb is the next option, giving us all kinds of byhad, onhad, withhad, tohad etc.
John had a good life on
Finally, we could make the subject be affected by the preposition of the object! This is somewhat natural - the spatial way we 'have' things is somewhat affected by the nature of the object. (Of course, for some combinations of object and subject, the combination itself may affect the manner of possession - maybe the language takes account of some such combinations, maybe not.) In this case, the language almost gets a peculiarly located split-alignment for possession. At this point we then get:
by Erin had a boatHistorically, the classes of possession-types may generalize into expressions for manners of having things, so that by changing the preposition, we change the implied relation of the possessor and possessum. Such semantic differences being explicit only in some TAMs is not unknown - the Georgian verb paradigm is a pretty clear example of markers that do distinguish meaning only appearing in part of the paradigm.
on John had a good life
to John had an idea
with me had a car
Now we're being a bit boring though - this distinction only happens with one particular verb (or cluster of verbs derived from it, if we consider the preposition a derivational device). So, how about we extend it to a bunch of verbs that are somewhat 'possessive-like'. We could include verbs that include physically moving things (carry, drag along, pull?, push?), verbs of allegiance and such, and maybe some other things as well. Suddenly we've extended the system a bit - and obviously, this system could permit for derivation in the same way it previously did. Of course, the derivation option is not what we're primarily interested in - that's just a door we open for derived languages.
* Of course s/pre/post if you want to, but I warn you, you'll run into the word postdicative and the phrase postsent tense if you do that to this post.