I want to be more precise.
About a sentence that expresses moral content, such as “Murder is immoral,” one can say that it is true or false.
But about a sentence that expresses a norm, such as “It is forbidden to murder,” there is no truth or falsehood here, because it depends on which system you obey, so here there is only binding or non-binding. Right? (Except that someone who holds that “Murder is immoral” will necessarily accept that “It is forbidden to murder” is binding.)

And one more thing: is there a connection between moral anti-realism and postmodernism? Everyone has his own narrative—there is no existing morality and everything is a social and personal construct.

I don’t see a difference between those two statements. “You shall not murder” is a command, and with regard to that there is no point discussing truth or falsehood.

You are not talking about anti-realism but about anti-objectivism. One can connect it to that, although regarding morality many people thought this way even before postmodernism.

So I didn’t understand the difference between realism/objectivism on the one hand, and anti-realism/relativism on the other.
If morality is real, then it is objective (and vice versa), and if it does not exist in reality (anti-realism), then it is subjective (relative), and vice versa.

I explained the difference above. If it is real then it is objective, but not vice versa.

According to moral realism, how does the fact that the object “murder” has the characteristic/property “bad” normatively obligate me not to commit murder? That’s the naturalistic fallacy, no?

I explained this. Ethical facts are not related to the naturalistic fallacy. Norms can indeed be derived from them. From the fact that it is forbidden to murder, one can derive that one must not murder. Beyond that, there is also the demand of the Holy One, blessed be He, to obey these facts.

Sorry for the nagging, but:
1. Above you wrote, “I explained the difference above. If it is real then it is objective, but not vice versa.” But you didn’t explain it, you only stated that there is a difference between them, without explaining what the difference is. So I didn’t understand the difference between them.
2. Here too you wrote, “I explained this,” but I don’t know where you explained it. You only stated that norms can be derived from ethical facts. But the question is: why??

1. In the first message here I explained it. Here is the quote:
Moral objectivism is the view that morality is objective and not relative. Moral realism is the view that morality exists in some sense.
From this you can also understand that being real entails being objective, but not vice versa.
2. Why not? That is exactly the difference between ethical facts and ordinary facts. The naturalistic fallacy speaks about ordinary facts. But an ethical fact is that murder is forbidden; its content says that there is a prohibition, that is, that there is a norm. Therefore here the norm can be derived from the fact.

Regarding objective and subjective: is the color of the table in front of me an objective fact or a subjective one (= each person according to how he sees the table; one may see it as black and another may see it as white)?

And regarding the topic of this thread, as a moral realist and objectivist, how do you justify the fact that there are so many disputes in morality? The morality of today is not the morality of yesterday, nor that of tomorrow.
If everything is objective, how are we supposed to decide the hard questions of ethics? For example, is polygamy moral? Is birth control moral?

This is the philosophers’ chestnut. We have no way whatsoever to check it. Moreover, it is possible that what I see when I look at the table is what you call hearing Beethoven’s Ninth Symphony, or some type of perception that is entirely unfamiliar to you. One thing is clear: there is no necessity that everyone sees the same thing. If a person were to connect the waves at the wavelength of red to the center of yellow in the brain, he would see them as yellow. There is nothing essential here.

As for morality, first of all there are not really that many disputes. They are at the margins. A very large majority of things are agreed upon, and that itself is an indication of objectivism. Beyond that, there is development over time in scientific knowledge too, and that does not mean science is subjective.