Find The Winning Move

I WAS on board nine for Battersea's latest London League Division One match, against Hackney.

Position after 26...Ba6-c8 in Spanton (163) - John Tennyson (177)

White's two-pawn edge gives a large advantage, but there is only one move that preserves it; other continuations largely dissipate the advantage or even lead to a loss.
Can you find the winning move?
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The greedy 27.Rxc6?? loses a piece to 27...Bf5+ 28.Kb2 Rd2+ 29.Ka3 Bxc3, when analysis engines reckon Black is clearly winning despite White having three pawns for the missing piece.
A passive move such as 27.Re3 is also problematic. Black's best line is not clear, but something like 27...Bh6 (27...Bf5+ is also good) 28.Re2 Bg4 (again, the immediate 28...Bf5+ is a promising alternative) 29.Rf2 (29.Bb6!?) Bf5+ 30.Kb2 Bg7, makes it difficult for White to coordinate his pieces and get the queenside pawns moving. In many of the lines I investigated with Stockfish9 and Komodo9, White had to give up the exchange to push home his advantage (not an easy thing to correctly time over the board).
Other retreats are even worse, eg 27.Re2? Bf5+ 28.Kb2 (28.Ne4?? Ra8 wins for Black) Rd3 29.Rc2 Rd1, with a position the engines calculate as dead equal, although both sides can carry on.
In the game, I played the correct 27.Bb6! One point is that swopping into opposite-coloured bishops by 27...Bxe6 28.Bxd8 Bxc3 does not give realistic drawing chances as Black's king is too far from the queenside. White quickly creates a pair of passed pawns and wins without much trouble.
So JT felt obliged to move his rook from the e file by 27...Rf8. There followed the cold-blooded 28.Rxc6 Bf5+ 29.Kd2, when White's king may look exposed at first glance, but is quite safe. The game saw 29...Bh6+ 30.Be3 Bg7 (or 30...Rd8+ 31.Nd5) 31.Nd5 Ra8 32.a4 (1-0, 41 moves).