I always try to prevent myself from saying "The Algarve" and just say "Algarve". "Al" is the Arabic definite article and the name Algarve comes from the Arabic "Al Ghareb" meaning "the west". So to me, saying "the Algarve" is like saying "the the west".
How many Brits I wonder go on holiday to Algarve having prepared a list of the freguesias in the area they are going to? Indeed how many Brits in Algarve have even heard the word freguesia?