Shift burden from income tax -> property tax. My country (Croatia) has a similar issue and the finance minister even said he thinks this is the correct move but there's no "political will" to do this.
I would increase property tax exponentially after the 5th property or so, to hit the big landlords and investment funds, while keeping the homeowners and small landlords appeased. Then take that money and build or buy public housing to rent for cheap prices, to bring the prices down for everyone.
Economic situation (and mentality) is very similar - Portugals problem seems to be more extreme though since it's located in a more interesting area (closer to developed Europe).