The truth is that legal immigration to the U.S. is difficult.
Laws are continuously being revised to make it even more difficult.

Visas can be classified into two broad types:

  1. Non-immigrant
  2. Immigrant

According to the Department of State: "There are more than 20 nonimmigrant visa types for people traveling to the United States temporarily.
There are many more types of immigrant visas for those coming to live permanently in the United States.
The type of visa you need is determined by the purpose of your intended travel."

Furthermore, immigrant visas can be subdivided into two main categories:

  1. Family sponsored
  2. Business-related

A resident alien (green card holder) or a U.S. citizen are both considered to be "U.S. persons".
Now, investment in real property is not the only way to acquire the U.S. person status.
There are others, for example a petition from a close relative who is a U.S. person, this would fall under the family-sponsored visa type.
Likewise, a qualified foreign employee or manager can become sponsored for immigration by a U.S. company, this would be an example of business-related visa.

This is not a comprehensive treatise on immigration, we will only deal specifically with how to leverage a purchase in real estate to achieve legal residence in the U.S.
For those interested, the most comprehensive resource is the one found in the above mentioned website.