ImplicitRegion[
  16 π^2 -
    Sqrt[32 π^2 + Abs[Cos[θ1] r1]^2 +
      Abs[r1 Sin[θ1]]^2] Sqrt[
     Abs[Cos[θ1] r1]^2 +
      Abs[Cos[φ1] r1 Sin[θ1]]^2 +
      Abs[r1 Sin[φ1] Sin[θ1]]^2] + Cos[θ1]^2 r1^2 +
     Cos[φ1] r1^2 Sin[θ1]^2 <= 0.5 &&
  16 π^2 -
    Sqrt[32 π^2 + Abs[Cos[θ1] r1]^2 +
      Abs[r1 Sin[θ1]]^2] Sqrt[
     Abs[Cos[θ1] r1]^2 +
      Abs[Cos[φ1] r1 Sin[θ1]]^2 +
      Abs[r1 Sin[φ1] Sin[θ1]]^2] + Cos[θ1]^2 r1^2 +
     Cos[φ1] r1^2 Sin[θ1]^2 >= -0.5 &&
  16 π^2 -
    Abs[r2] Sqrt[32 π^2 + Abs[r2]^2 +
        Abs[r1 Sin[φ1] Sin[θ1]]^2 +
        Abs[r1 Sin[θ1] - Cos[φ1] r1 Sin[θ1]]^2] +
    r2^2 <= 0.5 &&
  16 π^2 -
    Abs[r2] Sqrt[32 π^2 + Abs[r2]^2 +
        Abs[r1 Sin[φ1] Sin[θ1]]^2 +
        Abs[r1 Sin[θ1] - Cos[φ1] r1 Sin[θ1]]^2] +
    r2^2 >= -0.5 && 160 <= r1 <= 300 && 160 <= r2 <= 300 &&
  0 <= φ1 <= 2 π && 0 <= θ1 <= 2 π,
  {r1, r2, φ1, θ1}]

ImplicitRegion[
  -633.655 <=
  r1 (r1 (4. + 1. θ1^4 - 2. θ1^2 ϕ1^2) -
        Sqrt[(128 π^2 + r1^2 (4 + θ1^4)) (4 + θ1^4 + θ1^2 ϕ1^4)]) <=
  -629.655 &&
  314.827 + 2. r2^2 <=
  r2 Sqrt[128 π^2 + 4 r2^2 + r1^2 θ1^2 ϕ1^2 (4 + ϕ1^2)] &&
  r2 Sqrt[128 π^2 + 4 r2^2 + r1^2 θ1^2 ϕ1^2 (4 + ϕ1^2)] <=
  316.827 + 2 r2^2 &&
  r1 >= 0 && r2 >= 0 && 160 <= r1 <= 200 && 160 <= r2 <= 200 &&
  -0.1 <= θ1 <= 0.1 && -0.1 <= ϕ1 <= 0.1,
 {r1, r2, θ1, ϕ1}]