Regular quantum mechanics is linear, and as the linked question makes clear, that linearity is essential to prevent signaling given the rest of quantum mechanics: entanglement, non commuting operators, projection onto eigenspaces by strong measurements, etcetera.
So if you break the linearity and want to avoid signalling, then you have to change at least some of those other things. But there is strong evidence for all of those, and those properties are usually what we associate with quantum mechanics. So breaking them seems wrong since they have experimental evidence, and since they seem essential to what we call quantum mechanics it seems strange to call the result quantum mechanics if we do break them. The other issue is that I'm not sure what physics concept you want to know about (the linked post wanted to know about linearity and classical linearity versus quantum linearity). Your post might be about research or non-accepted physics and might be off topic, or maybe I'm not clearly picking up on your question.
Now, lets get to the answers. You could get rid of entanglement. After all, it is only linearity that made us think that entangled states are possible, you could just have every multiparticle state be factorizable. Then the burden is on you to explain experiments that used to use entanglements. This is actually possible, but the price can be high, for instance you can make a deterministic theory with conspiracy level coordination, that the same physical effects that determine which way a particle goes through a stern-gerlach machine also causally affects how we choose to orient the machine so as to produce the correlations we see when we make our supposedly-free-or-random-uncorrelated-choices. Any theory that postulates that our thoughts or our random number generators are compromised by a vast conspiracy is unlikely to be accepted without some serious convincing, so quantum theory will likely prevail, even if this theory were right in some sense. And there is room for some of these in accepted physics, some people research whether there is evidence we live in a simulation, and a conspiracy is easy to maintain if we live in a simulation, though again we in the simulation aren't going to find that very convincing since you can get into solipsisms, which are some other options, if the universe reached heat death but persisted for an incredibly long time, then there could be a statistical fluctuation large enough to form a Boltzmann brain, and if you (or I) am a Boltzmann brain, then the memories in the brain might have little to no relationship to actual experience. This approach is pressing up against the border of denying all experimental evidence of any kind. It's pretty bad. There are less bad options, you can have hidden variables for up and down in spin that are quite complex, designed to have enough degrees of freedom to explain what we've seen so far. Some people like to see if an experiment leaves any room for gaps, so you can comb through the existing experiments and look for gaps and loopholes to try to fit your entanglement-free theory into. And if your theories are credible, experimentalists can point to your theory as justification to spend time (and/or money) to do a better experiment. But most physicists aren't going to be surprised if your theory has to retreat further and further into new gaps as more experimental data comes out. So maybe there won't be interest and you'll just sit inside some gap forever because not enough people are concerned about your theory.
You can give up noncommuting operators, but most would say the result shouldn't be called quantum mechanics. But usually the operators in question are considered linear operators and you are getting rid of linearity, so why not address that? After all, we generate our sample spaces for mathematical probability theory out of maximal commuting subalgebras, so really noncomutting operators aren't technically ever tested, even in regular quantum mechanics. You can again have a superselection (not in the usual sense of the word) principle that generates maximal subalgebras of commuting operators and work with transitions between them. This doesn't seem much different than the previous case to me, you have to make a whole new theory that explains the results of the previous theory, in this case I'm actually saying you can try to confine yourself to the part that is most directly compared to experiment, and make a nonlinear version of just that (instead of making a nonlinear version of the whole theory superstructure).
Or you can instead attack projection onto eigenspaces. That's very promising, and again it's a linear operation. And since all real measurements are weak (to some degree) and all real measurements take time (and so are actually described by interaction hamiltonians) then we know the projection postulate is actually ever so technically wrong. But just because it is wrong doesn't mean that anything you replace it with is fine, some replacements might be good and some might be bad. But here I think you can find people that would study it. Simpler examples could be a nonlinear many worlds interpretation, where you engineer your lack of signalling by putting them in different worlds. To me it would smell like cheating, but I'm sure people would study it (if it hasn't been studied already).
