Restricting the domain to Reals will speed up the solution.

NSolve[{
  rp t1 - r (t1p - t2p) t3p + rp t4 - r (2 t1p + t2p) t4p == 0,
  t1p (r + t3) - rp t1 t3p - 2 t1p t4 - 2 rp t1 t4p == 0,
  -t2p t3 + rp t2 t3p - t2p t4 - rp t2 t4p == 0,
  -rp (t1p - t2p) t3 + (t1 - t2) t3p == 0,
  -rp (2 t1p + t2p) t4 + (r - 2 t1 - t2) t4p == 0,
  r^2 + rp^2 == 1,
  t1^2 + t1p^2 == 1,
  t2^2 + t2p^2 == 1,
  t3^2 + t3p^2 == 1,
  t4^2 + t4p^2 == 1,
  0 <= r <= 1,
  -1 <= t1 <= 1,
  -1 <= t2 <= 1,
  -1 <= t3 <= 1,
  -1 <= t4 <= 1,
  0 <= rp <= 1,
  -1 <= t1p <= 1,
  -1 <= t2p <= 1,
  -1 <= t3p <= 1,
  -1 <= t4p <= 1},
 {r, t1, t2, t3, t4, rp, t1p, t2p, t3p, t4p},
 Reals]

{{r -> 0, t1 -> 0, t2 -> 0, t3 -> 0, t4 -> 0, rp -> 1., t1p -> -1., t2p -> 1., t3p -> -1., t4p -> 1.}, {r -> 0, t1 -> 0, t2 -> 0, t3 -> 0, t4 -> 0, rp -> 1., t1p -> 1., t2p -> 1., t3p -> -1., t4p -> -1.}, {r -> 0, t1 -> 0, t2 -> 0, t3 -> 0, t4 -> 0, rp -> 1., t1p -> 1., t2p -> 1., t3p -> 1., t4p -> -1.}, {r -> 0, t1 -> 0, t2 -> 0, t3 -> 0, t4 -> 0, rp -> 1., t1p -> 1., t2p -> -1., t3p -> -1., t4p -> -1.}, {r -> 0, t1 -> 0, t2 -> 0, t3 -> 0, t4 -> 0, rp -> 1., t1p -> -1., t2p -> -1., t3p -> -1., t4p -> 1.}, {r -> 0, t1 -> 0, t2 -> 0, t3 -> 0, t4 -> 0, rp -> 1., t1p -> -1., t2p -> -1., t3p -> 1., t4p -> 1.}, {r -> 0, t1 -> 0, t2 -> 0, t3 -> 0, t4 -> 0, rp -> 1., t1p -> 1., t2p -> -1., t3p -> 1., t4p -> -1.}, {r -> 0, t1 -> 0, t2 -> 0, t3 -> 0, t4 -> 0, rp -> 1., t1p -> -1., t2p -> 1., t3p -> 1., t4p -> 1.}, {r -> 0.949484, t1 -> 0.181935, t2 -> 0.655977, t3 -> -0.655977, t4 -> 0.181935, rp -> 0.313817, t1p -> 0.983311, t2p -> -0.754781, t3p -> 0.754781, t4p -> -0.983311}, {r -> 0.949484, t1 -> 0.181935, t2 -> 0.655977, t3 -> -0.655977, t4 -> 0.181935, rp -> 0.313817, t1p -> -0.983311, t2p -> 0.754781, t3p -> -0.754781, t4p -> 0.983311}, {r -> 0, t1 -> 0.201794, t2 -> -0.897879, t3 -> -0.897879, t4 -> -0.201794, rp -> 1., t1p -> -0.979428, t2p -> -0.440243, t3p -> 0.440243, t4p -> 0.979428}, {r -> 0, t1 -> 0.201794, t2 -> -0.897879, t3 -> -0.897879, t4 -> -0.201794, rp -> 1., t1p -> 0.979428, t2p -> 0.440243, t3p -> -0.440243, t4p -> -0.979428}, {r -> 0, t1 -> -0.201794, t2 -> 0.897879, t3 -> 0.897879, t4 -> 0.201794, rp -> 1., t1p -> -0.979428, t2p -> -0.440243, t3p -> 0.440243, t4p -> 0.979428}, {r -> 0, t1 -> -0.201794, t2 -> 0.897879, t3 -> 0.897879, t4 -> 0.201794, rp -> 1., t1p -> 0.979428, t2p -> 0.440243, t3p -> -0.440243, t4p -> -0.979428}, {r -> 0.196601, t1 -> 0.251666, t2 -> -0.907394, t3 -> 0.907394, t4 -> 0.251666, rp -> 0.980484, t1p -> -0.967814, t2p -> -0.420282, t3p -> -0.420282, t4p -> -0.967814}, {r -> 0.196601, t1 -> 0.251666, t2 -> -0.907394, t3 -> 0.907394, t4 -> 0.251666, rp -> 0.980484, t1p -> 0.967814, t2p -> 0.420282, t3p -> 0.420282, t4p -> 0.967814}, {r -> 1., t1 -> -1., t2 -> 1., t3 -> -1., t4 -> -1., rp -> 0, t1p -> 0, t2p -> 0, t3p -> 0, t4p -> 0}, {r -> 0, t1 -> -0.979428, t2 -> -0.440243, t3 -> -0.440243, t4 -> 0.979428, rp -> 1., t1p -> 0.201794, t2p -> -0.897879, t3p -> 0.897879, t4p -> -0.201794}, {r -> 0, t1 -> 0.979428, t2 -> 0.440243, t3 -> 0.440243, t4 -> -0.979428, rp -> 1., t1p -> -0.201794, t2p -> 0.897879, t3p -> -0.897879, t4p -> 0.201794}, {r -> 0, t1 -> 0.979428, t2 -> 0.440243, t3 -> 0.440243, t4 -> -0.979428, rp -> 1., t1p -> 0.201794, t2p -> -0.897879, t3p -> 0.897879, t4p -> -0.201794}, {r -> 0, t1 -> -0.979428, t2 -> -0.440243, t3 -> -0.440243, t4 -> 0.979428, rp -> 1., t1p -> -0.201794, t2p -> 0.897879, t3p -> -0.897879, t4p -> 0.201794}}