Bone Marrow Transplant DNA mutations

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Serious question. Please no troll answers.

Recently I learned that you can't do a 23 and me of you have had a bone marrow transplant before. The reason being that your bone marrow is the organ that creates blood cells, and since you have someone else's bone marrow, you will have foreign blood DNA and can no longer DNA testing.

My question is this: Going forward after a bone marrow transplant, will your body start to incorporate this new DNA? Could your DNA going forward be completely changed?