It's going to be hard for any position to be "natural" if the king is out and roaming about, so I presume that you mean that the position must be legal. Also, as far as I understand it, the king may only teleport in a checkmate position, and not while in check, because otherwise, it would be nigh impossible to give a mate.
Firstly, it is entirely possible to get eight queens on a board for checkmate without drawing, like so, with a little help from other White pieces. While this can be done with just the pawns alone, the extra pieces are needed to do it in exactly eight moves.
[FEN "8/PPPPPPPP/8/8/8/8/k7/1RN1K3 w - - 0 1"]
1. a8=Q+ Kxb1 2. c8=Q Kb2 3. d8=Q Kb1 4. e8=Q Kb2 5. f8=Q Kb1 6. g8=Q Kb2 7. h8=Q+ Kb1 8. b8=Q#
But it is also possible to achieve this effect with just White's eight starting pieces. I remembered this old problem by Josef Kling that does it. The stipulation reads: White checkmates in 14 moves but only after attacking all 64 squares." I would suppose that this position is more "natural" than one with eight White queens.
Do note that while the stipulation reads “64 squares,” it has been proven, that with opposite-colored bishops, the starting pieces can only control 63 squares. Kling knew that for sure, so it's just a minor discrepancy that he used “64” to have a better sounding stipulation in my opinion.
[Title "Josef Kling, The Chess Player's Companion 1849, 'Fancy Sketch'"]
[FEN "4k3/8/8/8/8/8/8/RNBQKBNR w - - 0 1"]
1. Qd6 Kf7 2. Rh8 Kg7 3. Bb2+ Kf7 4. Qa6 Ke7 5. Bh3 Kf7 6. Ke2 Ke7 7. Kd3 Kf7 8. Bd4 Ke7 9. Nd2 Kf7 10. Ne2 Ke7 11. Rg1 Kf7 12. Nc3 Ke7 13. Nd5+ Kf7 14. Be6#