Before we embark on an analysis of the verses from the Quran related to our topic, it is also helpful to look at some proxies that are already before our eyes. What do I mean? There are some issues that are controversial today but were not when the Quran was revealed, such as polygyny and slavery, and Islam did not abolish them but regulated them. On the other hand, alcohol consumption and interest was widespread among Arabs when Islam came, yet Islam abolished them. Coming to our topic, polyandry is generally a controversial topic, although there are a few communities around the world who practice it. Still, we can hypothetically ask "if there had been polyandry in the Arabic society of the 7th century, would Islam abolish it or regulate it?"

Let's look at polygyny, i.e. one man multiple wives. When Islam came, there was no limit to the number of wives a man could take. There was no ethical regulation in regards to the way women were treated in the family, either. With the relevant verses in the Quran, three things happened. One, a limit to the maximum number of wives was decreed. Two, under what type of conditions this permission could be used was told. Three, the relationship between the spouses was established on mutual love, respect, and fear of God. This last item about fear of God may sound strange to you, but it means that one should be considerate of the fact that they are going to be held accountable for their treatment of their spouse on the Judgment Day. So, the loving atmosphere of the family becomes a manifestation of the love of God, whereas restraining one's anger and avoiding excessive behaviors towards each other becomes a manifestation of fear of God. In short, Islam not only regulated polygyny, but also improved the family life in general.
Today, although there are communities practicing polygyny, most people consider it as unacceptable. Even the Muslim ladies have a hard time in internalizing the fact that their holy book contains such a permission. Families in a polygyny setting are met with distance even among the Muslims. Tragic thing is people consider cheating as a normal guilt, but polygyny as an out-of-question practice. Yet, polygyny is there in the Quran, no matter what kind of a context you put it in. I am not going to question this psychology, because it has been partially addressed in the previous episode and in another article and because my main goal here is to pave a road towards understanding the Quran. Accordingly, the question is "although, looking from today, we would like Islam to abolish polygyny completely, why did it not do so but sufficed with regulating it?"

When we start questioning the commands and prohibitions of God, the first thing we must remember and keep in mind is that there doesn't have to be a wisdom or reason for any of such instructions. If God wants it, He has the authority to command it. However, once beyond that point, we can investigate the wisdom behind such orders, because Allah is All-Wise, and there is wisdom in all His actions.
With this perspective, when we look at family, society and the mutual roles and responsibilities among the spouses, we see that family is positioned as part of a social security system in Islam. Therefore, abolishing polygyny means breaking this social security system where women and children are going to suffer due to lack of support both economically and spiritually. That break down is destined to have consequences in the moral state of the society, as well. So, institutionalizing a state-regulated social security system can for sure undertake the economical aspect of the issue, but the proper family atmosphere cannot be established only with money.
Here, I am not trying to defend or offend one type of application, but I am just trying to figure out some of the wisdom behind the permission of polygyny in Islam. And in doing so, instead of trying to look at it through our eyes, I am trying to emphatize with the context set in Islam. And we see that permission of polygyny is linked to avoiding other prohibitions and to fostering the human psyche and body as a whole so that both individually and collectively people can be steadfast on the way of Allah.

Slavery is another proxy we are going to look at. After the dire memories recorded in history, slavery is appalling and cannot even be thought of as an institution in a society. When Islam came to the 7th century Middle East, though, slavery was widespread not only among Arabs but also all around the world. It was both part of the economy and an integral part of the social security. However, slavery is slavery, and people who fell into its grasp could be treated in ways that are not fit for the human honor. When Islam came, although it encouraged freeing the slaves and elevated their social and economic conditions, did not abolish the institution. That is, instead of abolishing it, Islam regulated it.
Again taking on the same respectful attitude, as believers, we accept this situation, but we are free to question and analyze the wisdom behind it so that we can project the same wisdom to other cases if needed. For example today, although we reject even the idea of slavery, the state of the people in the modern work life and environment, the treatment of the working class by the wealthy elite is not too different than the state of the slaves back then. So, modern regulations for taxation and employee rights can benefit from the wisdom that shines from the framework outlined in the Quran considering the slaves. To start with, for example, the slaves are referred to with different names according to the context. Such variable naming can be interpreted as a sensitivity towards the standards of fair treatment for the involved people, because they are still honorable human beings, not perishable commodities. Therefore, from Quran's point of view, slavery is not slavery, but a manifestation of a fundamental component of human social life, albeit one that needs regulation and control.
So, with the proper paradigms in mind, we see that the Quran did not abolish slavery, because it is a projection of a timeless paradigm onto a society in history. Prohibiting such a thing would be against the nature, because the existence of both rich and poor or influential and unable in a society is God's decree in the first place. This polar state of the affairs necessitates give and take between those poles, which needs to be regulated not blocked.
There are other topics, but to save the extensive discussion, I am just going to mention them and give the gist of the matter that can help us on the way to analyze the situation with polyandry.
Interest, unlike the two discussed above, is abolished and prohibited explicitly in Islam, although it was widespread among Arabs back then and was part of the economic structure. If you look at it, you see that what is forbidden is gaining without a corresponding work or investment. So, preserving the value of human labor and avoiding the formation of a class that thrives on the work of others is in the forefront. It is this reason that modern Islamic thinkers talk about accepting interest if it is at the same level of inflation, because what is done is the preservation of the value of human labor even if under the label of interest.
Waging wars is an unfortunate but undeniable part of humanity. Claiming to be a religion of peace, Islam did not prohibit wars, but regulated them. Here, again a recognition of the human nature in its entirety and an effort for the protection of the essential values (life, freedom, etc.) is visible. This is how we can distinguish the terrorists, who claim to be fighting in the name of Islam, from those who fight for a true cause. The formers are serving to the formation of a state of affairs where human life is under constant threat and freedoms are inherently violated, whereas the latters are serving to the establishment of a structure that fosters the flourishing of the mentioned rights and values.
Consuming intoxicating materials is another prohibition is Islam. As today, in the 7th century Arabian Peninsula, too, alcohol was used by virtually all. Yet, Islam forbade its use. When we look at the wisdom behind this prohibition, we first see that there are health issues due to alcohol consumption. On top of them, though, suppression of conscience and reason is another detrimental aspect, because one of the aims of Islam is to orient people to a higher level of conscience and more effective use of reason.

After these clearly forbidden, commanded or permitted matters, I would like to bring two other issues that are not explicitly mentioned in the Quran, but were forbidden during the initial times of Islam among the Arabs. The first is idol making. In modern terms, sculpture. Although there is not a single verse in the Quran in this regard, prophet Muhammad (pbuh) strictly prohibited the making of statues and sculptures. Although a significant portion of the Muslims until recent times, and some still, took this as an unconditional ban on sculpture, we today see that this attitude by the messenger of God was there to avoid any triggers of idol worshipping in a society that had idolatry genes in its culture. In order to fortify the unity of God both in minds and in the daily life, old applications linked to idol worshipping had to be abandoned. However, today, we are away from those conditions, and there is no point in continuing the same attitude.
A contrasting issue with the above is the mut'ah. This "temporary marriage" was the norm when needed among the Arabs at the time. However, it is clear that it is a form of prostitution. Some people today claim that it is a form of marriage and so, it is allowed in the context of Islam. However, the framework for marriage drawn in the Quran involves an intention to be together forever. So by definition, Quran outlaws mut'ah, even if it is not spelled out in a verse. Accordingly, prophet Muhammad prohibited it. Therefore, no matter what the conditons are, one cannot talk about mut'ah as a valid form of marriage, because it violates the framework of marriage drawn in the Quran.
Obviously it is impossible to find everything in the Quran in a spelled-out manner. Still, we can use our mind and extract the essence from what is provided to us, and thus, we can make an educated guess on what is pleasing to Allah in the new situations that are not listed openly in the Quran.
Therefore, considering all of the above, if there had been polyandry in the Arabic society of the 7th century, would Islam abolish it or regulate it? I think it is safe to say that if there is not a verse that openly prohibits polyandry, if the application of polyandry does not involve or lead to the violation of the principles and values set by the Quran, Islam would either not say anything about it or regulate it. And so, if we confirm these from the Quran, we, too, must stay tolerant towards the idea, whether we like it or not.