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Eutectic systems behavior near 100%-0% composition and low temperature
As you cool down into the two phase region the A-rich crystal can no longer hold the B in solution. The B will start to diffuse out of the A rich crystals and form new B rich crystals. This will normally happen at the grain boundaries and so you end up with lots of tiny B-rich crystals around the larger A-rich crystals. This does indeed all happen in the solid phase. Even in a solid, atoms can diffuse around the lattice. But if you cool faster than the atoms can diffuse you can force a material into a single phase. But then the standard phase diagram in the original question would not apply.