I was reading Barry Mazur's biography and come across this part:
Grothendieck was exceptionally patient with me, for when we first met I knew next to nothing about algebra. In one of his first conversations with me, he raised the question (asked of him by Washnitzer) of whether a smooth proper algebraic variety defined over a real quadratic field could yield topologically different differentiable manifolds realized by the two possible imbeddings of the number field into the reals. What a perfect question, at least for me! Not that I answered it. But it was surely one of the very few algebro-geometric questions that I then had the background to appreciate. ... the question provided quite an incentive for a topologist to look at algebraic geometry. I began to learn the elements of algebraic geometry working with Mike Artin.
Is the problem still open? I am an algebraic topology student so it feels very surprising someone will come up with a question like this. But I am at a loss how to experimentally find some toy examples one can work by hand.