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I'm planning a trip to Aruba and so I was checking out the passport requirements. I found them here. Near the top, it says, "During their stay in Aruba tourists are not allowed to work" (emphasis theirs)

Does this mean that I can't so much as check my work email? And if so, when I tell my employer about my travel plans, should I mention that if they ask me to do any remote work while I'm there, they'll be asking me to violate the law? I want to use this as an excuse to "turn off" while I'm there.

EDIT: Part of the reason I'm asking is because of some forum posts I've found where there isn't a consensus about this being allowed. As for my employer asking me to work remotely, for my job that just means being available in case anything goes wrong and answering emails if anyone needs an answer urgently.

EDIT 2: I'm a US citizen.

The link you cited states the following:

Persons who are considered a tourist are those who travel to Aruba for one of the following purposes: vacation and relaxation, sport, health reasons, family matters, study, religious purposes or a business visit. During their stay in Aruba tourists are not allowed to work.

So, the "not allowed to work" part definitely isn't excluding working remotely for your normal (foreign) job while briefly visiting Aruba, even if you're actually conducting business meetings with clients or vendors in Aruba. Odds are your employer knows this, as it's the same in almost every country, so I wouldn't recommend attempting to use that as an excuse to your boss.

The "not allowed to work" part in most countries means that you can't get a job working for someone in that country or start a business in that country, etc. while you're there. This is aimed primarily at illegal immigrants circumventing the normal work visa requirements and taxes. While the technical legal definitions often don't define this carve-out well, in reality, even if they do know, no one is going to care that you answered some work e-mails or even did some design work in the evenings during your visit. As long as you can truthfully tell the immigration officer that the purpose of your visit is tourism/vacation, you're good and there's normally no reason to even mention checking e-mails or similar to them.

There are two issues here:

1. "not allowed to work"

This refers (normally) to taking [illegal] employment in-country on a tourist visa. And thus avoiding the whole employment visa process, and paying taxes. FYI, here's how Aruba defines work.

Generally speaking, Immigration doesn't care about you checking your work email. They're usually more concerned about illegal work and immigration.

2. "an excuse to turn off while I'm there"

You don't need an excuse. You're on vacation. Depending on where you're from, pestering you to check your emails and do some work could be a (serious) violation of your country's laws. If you want to turn off while on vacation, leave your work phone and laptop at home.

Using I want to use this as an excuse to "turn off" while I'm there. is beyond ridiculous. Your manager would probably laugh at you so hard your ears would ring even after your vacation is over. Don't. If you need an excuse to not answer emails, you need to find another job.

De jure there are dozens of complex laws and regulations regarding remote employment, under which you may or may not need a special visa in order to work in a given country. Tax laws are an additional complication, where countries such as the UK can deem you as a tax resident for spending as little as 16 days on British soil.

De facto, as long as you don't mention your remote job to immigration personnel at the airport, there's a 99.99% chance no one will ever find out. There are millions of people breaking the law by being employed at on-site jobs in any given country, so remote workers who travel for short periods of time are a pretty low priority for law enforcement.