I think Rxe4 is the only move which avoids checkmate or queen loss. However after Rxe4 Qxe4 black has a winning position as his rooks are very strong and white has no counterplay. Black pawns in the centre are also mighty and it is not that difficult to create a passer
White pieces are too far away to defend king and passive knights are much weaker than a rook. So despite it might seem white is going to be only a pawn down (when simply counting material) black's positional advantage is really great
White played Qe3 in this position (which resulted in mate in 2). Would Re4+ have been a better move? Or maybe something else? And which color would win this battle in the end? White has more value but lacks position. I'm interested in your opinions!