this is a devilish problem, it's like something out of a Dvoretsky book. Not knowing the outcome gives it an edge.
If Black plays 1... Kg6 then I think the best he can do is draw, and if he's not careful he could lose.
1... Kg6 2.Rxa3 Rh8+(Rxa3 draws) 3.Kg3 Kxg5,
and then 4.Ra6 Rh4 5.Rxd6 Rxg4+ 6.Kf3 Kf5 7.Re8 Rf3+ 8.Kg3 Re4 draws.
Instead, 4.Ra7 gives Black a bigger headache, the rook's heading for e7 and Black's forced to choose between passive defence or counterattack.
He can't go for the g4 pawn right away,
4... Rh4? 5.Rg7+ Kh6 6.Kxh4 Kxg7 7.Kh5 wins, and trying to defend the pawn doesn't offer much hope either,
4... Rh6? 5.Rg7+ Rg6 6.Rxg6+ Kxg6 7.Kh5 wins, and
4... Re8?! 5.Rg7+ Kh6 6.Rf7 Kg6 7.Rf6+ Kg5 8.Kf3 Re7 9.Rf8 Rg7 10.Rh8 Rf7+ 11.Kg3 Rg7 12.Rh5+ Kg6 13.Kh4 Kf7 14.Rh6 wins.
I think the best try for Black is 4... Rc8(or a8/b8) and then
5.Rg7+ Kh6 6.Rf7 Kg6 7.Rf6+ Kg5 8.Rxe6 Rd3+ 9.Kf2 Kxg4 draws.
So if Black plays 1... Kg6 and White offers the draw with Rxa3, then Black saves himself a lot of bother accepting the draw.
I think that in the original position Black has the edge, only the white king is available to protect the white pawn islands...*ponders*