One of the most controversial questions is hard to answer and that's why it's so controversial. Firstly, this isn't legal advice.

Remote work is such a recent and new technological development and governments are still lagging behind and figuring out how to deal with it. It's still quite a small niche compared to the overall population: Berlin might have thousands of nomads, but it also has close to 4 million people, so it's ~0.1% of the population.

Generally nobody is allowed to work in any country on a tourist visa, that's why there's work visas/permits. But this law is based on old times where work happened in offices and factories. What if an American working for a U.S. startup checks and answers their work email in a Berlin hotel while visiting on a tourist visa? That'd constitute work in official legal terms, but should that be illegal? Probably not.

Work permits are there to protect the local population from having anyone come to the country and compete with them on getting jobs. This makes perfect sense as you don't want the whole world to just come fly to your country and start working and pushing wages down for locals. However, 99% or more of remote workers visiting a country don't compete with locals on jobs. They're not going to restaurant kitchens to cook, instead they'll be in Berlin coding for startups in San Francisco which has customers from all over the world. If anything, governments would want to promote that as it means more consumer spending than tourists, and socializing with locals might mean technological, knowledge and cultural exchange, not at all increased competition to local workers.

However, that doesn't mean it's legal. Most countries have not spoken out about the legality of remote workers visiting their country and working without a work permit, and the ones who have have mostly made comments that it's okay if it's not work competing with locals, but those aren't official declarations. If you're strict, you can say it's illegal. If you're less strict, you can say it's a legal gray area due to laws lagging behind the reality of remote work. Personally, we believe the last. There haven't been any known cases of people getting arrested in foreign countries for working remotely, yet. In this case, Europeans have it easy, they can legally work anywhere in the European Union without any trouble.

It's therefore completely up to you and your risk assessment to choose what to do. Getting a work permit is possible in many countries, although it takes a lot of effort, tax constructions and lawyer fees, and it implies you're going to stay there for many years, which many remote workers aren't sure of. That makes it complicated.