Doug here again.
So long as the trust documents allow the trustee to use assets of the trust to acquire real property---whether the trust be a revocable trust (living trust) or an irrevocable trust, yes, trusts may purchase real estate legally.
Because the trust is the owner of the property though, if there is lawsuit against the owners of the property for some reason---such as a suit by a tenant or someone who is injured on the property and sues for negligence----then yes, the trust assets would be placed at risk if there was insufficient insurance to cover any liability.
Also keep in mind that property/real estate must be maintained and maintenance would need to come from the trust assets which could prove problematic in the event of a significant repair. And with real property, keep in mind that any of the trustees could at any time sue the other trusts for Partition and this would result in legal expenses to the trust.
So, while it is legal---it may not be the best idea in the long run for 3 separate trusts to buy property together.
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