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You would only need to pay taxes if you had capital gains from the REIT. For example, if you paid $50,000 for your interest in the REIT, and you sold it for $75,000, then you had $25,000 in capital gains, and you'd need to pay taxes on that $25,000. On the other hand, if you sold your interest in the REIT for $50,000, then you'd pay no taxes at all because you had no capital gains.
As an aside, the fact that you'd use the money to pay down a mortgage
on rental property is not a factor as to whether or not you'd pay taxes.
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