I depends on what you want your logic to model. You might want a logic that applies to some everyday process, like the laws, and not only to 'perfect' situations.
One way to look at this is in terms of types of modality. Modalities of obligation or of preference do not enforce non-contradiction. If you create a contradiction of obligations, there may be unavoidable loss, but the case is not impossible. if you create a contradiction of wishes you may feel guilty or torn, but life goes on. (Sorry to mention modals in every second post. I find this a productive unifying concept.)
Law is a modality of obligation, full of examples of contradictions that exist until they are resolved. So it is a good source of examples of violation of the law of non-contradiction.
Some of these contradictions are extremely explicit. The concept of common-law marriage is an example.
For a very long time in the U.S., into the 21st century in my own state of residence, (Illinois) an ongoing heterosexual relationship outside of marriage that also involved living in rental property rented in common was defined as unrelated co-habitation, and was illegal, as a form of fraud. (Even if you did not lie to the landlord.) But if you sustained such a relationship for seven years, you became married by common law, rendering the entire seven years worth of illegal behavior legal in retrospect.
This is a state of paraconsistency much like quantum superposition. Both the illegality of the situation and its potential future retroactive legality are real and opposite states that the current relationship is in.
Squatter's rights, considerations of easement, some kinds of custody, and other aspects of law have a similar status -- you sue someone to make existing illegal activity legal in retrospect and to place legal requirements upon the person whose rights you have officially been violating. And people cannot interfere just to prevent you from making this happen. If they want to maintain their rights, they have to be motivated by something genuine.
The law of the excluded middle also does not apply in this domain. You may enter a contract to provide some product, under the condition otherwise you must return some sum of money paid for it. And then your business might burn down, destroying that property and leaving you bankrupt. So you might do neither, while still not being considered to have violated the contract. We constantly make promises that one or the other thing will happen, which violate the law of the excluded middle in the case of our death or other extreme failures. It is not true that you either honor the agreement or violate it, leaving no other outcome.
The two are complements of one another, but they are not the same. In the language of constraint programming, they are opposite faults.
Contradiction happens when a situation is 'over-determined', when there are too many competing agendas involved to clean up the agreements ahead of time. Theoretically, then all outcomes are impossible. But there simply cannot be "no outcome" in real life. We just accept the one that happens and try to patch up history accordingly.
An excluded middle event happens when a situation is under-determined, when the objects involved are incomplete or imperfect by nature and their real behavior cannot be adequately foreseen. We just accept that none-of-the-above is sometimes forced upon us, and try to unwind the expectations in some balanced way.