Here is my experience. I have a primary Wyoming LLC with foreign LLCs in the different states I work in. I also own many rental properties in Wyoming, over 25, each of those properties has its own LLC and its own bank account.
I wanted to start doing business in New York state, and felt its ownership paperwork cumbersome, so since I already owned all those LLCs, and already having them as an expense, I created a new New York LLC (not a foreign LLC from my primary company) with each of the real estate LLCs owning an equal share of under 4% plus my primary LLC, I was able to avoid that paperwork and more importantly, avoid identifying myself to authorities. The law firm had zero issues in receiving funds from many real estate companies, helping with forming a new LLC, transferring funds to the new bank account (Citi), and helping hire a local manager with 0% ownership so keeping them out of the public records. Later, due to several reasons, I have had the NY LLC open several different bank accounts with various banks. Only Citi and Bank of America are the larger banks, the others are either local or regional. I had some initial issues with Citi at first but the law firm was able to resolve them painlessly.
When I wanted to expand to England and Wales but desperately wanted to avoid my name publicized in Companies House, I used my NY LLC to be the owner of the new company based outside of London. Since no UBO could be identified with any significant share of ownership, completely confirmed with the authorities in New York, my name was left off the public and private records. Citi had zero problem with allowing us to open a separate account for establishing this business and we hired a local director (actually a personal friend of mine who knows the industry) with 0% ownership and later a high street bank account (in addition to Citi). This director runs my business for me. Periodically I do travel to London but I am a "consultant" for my projects there. No one thinks I am the decision maker, but honestly I am not, I leave that to my director, he is that capable.
Not 100% legal in the UK but my privacy was much more important than keeping on the up and up. The interesting thing is NY laws are contradictory and I think I am legal there too.
I would not recommend doing this with anyone else unless you absolutely want to remain anonymous and have the revenues to justify it.