How can an F-1 student open an LLC and invest in real estate?

Anyone can invest in real estate in the U.S. You can do this as long as it is a passive investment. You would not be allowed to work for your real estate company or investments.

As a non-resident, there is no limitation in forming a business. Just as well, there is no limitation in purchasing and owning real estate as a non-resident.

You can own real estate and during your first period of practical training, you can arguably be self-employed if this relates to your area of study. If not best to appoint a property manager and limit your involvement to passive investing. Alternatively, the option is to obtain a green card or a work visa that clearly authorizes these activities.

It should be fine provided you are not employed there.

Ownership in companies, such as in the stock market, is allowed. But a foreign student will need to make sure that they are not engaged in any labor or that could cause problems for the student. I recommend you speak with a corporate immigration law specialist.

The managing of the business would likely be considered employment. A purely passive investment should be OK though.

You can open an LLC and be a passive investor, buying/holding/selling real estate; you just cannot work or receive compensation for services that would normally be performed by an employee.

Ownership of an LLC and passive investment income generally does not violate terms of an F visa.

It does not sound at all like a passive investment to me. If it is a passive investment, you simply send money to the company in which you are investing.