Given integers m and c satisfying m-2 >= c >= 2, we explicitly construct a nonsingular m-dimensional algebraic subset of P^{m+c}(R) that is not isotopic to the set of real points of any nonsingular complex algebraic subset of P^{m+c}(C) defined over R. The first examples of this type were obtained by Akbulut and King in a more complicated and nonconstructive way, and only for certain large integers m and c.