For certain kinds of congruence, there are two main types we can consider:
- morphosyntactic congruence
- semantic congruence
The former agrees in case, gender, etc, the latter agrees in some way with the meaning. We find English having a split on this with regards to examples like 'the family is' vs. 'the family are'. Different speaker communities do not agree on which one of these are right, and some may even use both with some subtle meaning differences. However, let's make up some more interesting thing here, such as a rule that tells us when the congruence is semantic and when it's morphosyntactical.
One idea could be that NP-internally, agreement always is morphosyntactical. (I will go and revise this later with regards to participles!) We could also go and say that VP-internally, the agreement is always semantic. However, the subject not being part of the VP, subject agreement on the verb might be exceptional - I'll go with morphosyntactical here. (Here, I am rather agnostic as to which way is most likely in a natural human language; heck, I find myself conflicted on whichever way VP-internal or NP-internal is more likely to go). For the language I am envisioning, the verb also has object congruence.
So, now we have a system where
the family sold-subj:3sg-obj:3pl the flock
We may of course have some gender congruence adhering to this pattern. Now, we may also have a complication with regards to quirky subjects and objects, or oblique ones: the subject might get a third person ('neutral') marking regardless of the subject's person, number or gender, while a primary 'semantic' object that is marked obliquely in some sense still might get some form of object marking on the verb.
Another complication we can introduce is with regards to left-dislocated objects - regardless if it's due to focus fronting or topicalization or whatever other thing, the object may then be considered outside of the VP, and the gap left behind now might not cause congruence.
Participles obviously have features both of adjectives and of verbs. There, passive participles could take semantic agreement, active participles morphosyntactical agreement.
Here, however, we get a lot of possibilities for sliding scales of marking, and this whole notion could be a nice testing ground for looking at probabilistic approaches for grammars: maybe, just maybe, we could build a system where the probability in some context for one kind of congruence is P, and for the other it's 1-P, where P is a real number in the range [0,1]. In different circumstances, the probabilities differ: subject marking on finite verbs has probability Psubj, object marking has Pobj, a left-dislocated object has Pl-obj, adjectives in attributive position have Pattr, and so on. These may further have a hierarchy where a change in the probability of one might force the probability of another to change.