Women Or Tigers? — The Third Challenge

‘The first two days of my new scheme have been a fiasco!’, the king said to his minister. ‘Both prisoners solved the logical problem I had set up for them. I will make the challenges more difficult!’
‘Splendid idea!’, said the minister.

On this day, as it was a public holiday, two prisoners had to face the challenge. They were both given the instruction that if there was a woman behind door #1, the sign on the door told the truth — but if there was a tiger behind it, the sign told a lie. As for the sign on door #2, it was the opposite: If there was a woman behind it, the sign would be false, and if there was a tiger behind it, the sign would be true.

There was still the possibility that both rooms could contain a woman, or both a tiger, or of course there might be a woman in one of the rooms and a tiger in the other.

The first prisoner was shown two doors with the following signs:

ROOM I: THERE IS A WOMAN IN BOTH ROOMS

ROOM II: THERE IS A WOMAN IN BOTH ROOMS

So which door should the prisoner choose?