When you regain control over Big Boss, you'll have to carry Quiet to a safe place without being able to use weapons. A sandstorm makes the whole process easier, but there are reinforcements coming towards the Lamar Khaate Palace. Walk south until you reach the rocks, then turn southeast. Stay close to the rocks and keep crouching. You can rely on the recon team and stop whenever there is a warning on the map (red circles). Head towards the southern marker where the game will play a cutscene.
After seeing the summary and new cutscenes, there will be a short epilogue. Don't exit the area of the mission, start following the footprints instead. They will lead Big Boss to the tree shown above with a cassette tape left by Quiet. Picking it up will result in achieving one of the secondary goals.