Let F be a real quadratic field in which a fixed prime p is inert, and E0 be an imaginary quadratic field in which p splits; put E=E0F. Let X be the fiber over Fp2 of the Shimura variety for G(U(1,n−1)×U(n−1,1)) with hyperspecial level structure at p for some integer n≥2. We show that under some genericity conditions the middle-dimensional Tate classes of X are generated by the irreducible components of its supersingular locus. We also discuss a general conjecture regarding special cycles on the special fibers of unitary Shimura varieties, and on their relation to Newton stratification.