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In the UK you can do it via Companies House for £15: https://ewf.companieshouse.gov.uk/runpage?page=welcome

Edit: Not sure about for foreigners though.

Yes, you can, both EU residents and out of EU residents.

But you need an official address in UK (registered address) for your company, where Government can send you official mail. There are companies that give you this service, so it's not a big problem.

"There are companies that give you this service, so it's not a big problem.", can you list some companies who provide this kind of service?

I registered my UK company while I was living in Germany. I used these guys for my registered company address: https://www.smallfirmsservices.com/Default.aspx

I used their cheapest service (£80/year) for which you get the company address, and they will only forward mail from HMRC and Companies House - anything else gets returned to sender.

We still use them, even though we have a trading address now.

How easy it was to get a bank account?

It was ok for me as I already had a UK domestic account and a UK address I could use, but I think from outside the UK this would be very difficult. I suspect your best bet would be HSBC as they operate an "international" model (supposedly).

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