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Are you trying to invest in local stock exchanges?   or international ones?

Local should be no issue and you can go through brokers.

International exchanges more problematic but there are services depending on how your financials are set up.

I am looking at US and UK markets which is where the difficulty is

I have it set up through HSBC share dealing account as I have a bank account with HSBC °µÍø½ûÇø offshore.  Local banks won't help with that.  There are a few offshore services which can potentially do that but are severely restricted.  Worst case, you have to open a bank account in Jersey or Isle of Man to be able to do this.

You are lucky to have HSBC offshore, they rejected Saudi and Bahraini residency status from their compliance department!

Look at iG, Interactive Brokers, Plus500, FreeTrade is going to be good as no fees (set to launch this quarter), you have to register now :

Most I have dealt with are fine with °µÍø½ûÇøs and registering online, however for compliance and money laundering regulations you need to be able to send them proof of identity and residence. This is normal.

Also, X-O do low cost execution only for UK stocks. For UK+US use Interactive Investor which took over the Natwest system, they have really good rates if you are a frequent investor.

I have had HSBC offshore since 2014 when I was in Dubai and had Dubai residency.  Strange that they don't accept Bahrain/Saudi residency. They only offer premier for certain nationalities though.