Hi, thanks for your question. My name's XXXXX XXXXX I'm going to assist you with it this afternoon.
As you correctly point out, the relevant area of law here is "passing off". A legal right to use the name does not arise purely because you have registered it at Companies House but the other business may be liable to you under the laws of passing off if, by using the name, they confuse customers (whether intentionally or not), exploit your established goodwill and you suffer a loss of business as a result.
Passing off is essentially there to protect the established reputation of a business and, in order to bring a successful claim, you would need evidence of confusion and that this has affected the goodwill of the business. You are clearly in the same trade but, in order to establish "confusion", you will need to be take all relevant facts into account. These include matters such as location (clearly, if one business services Brighton and the other, Liverpool, customer confusion is unlikely to occur).
If you want to address this with them, send a letter identifying yourself and your business, drawing their attention to your registered company name and that you have established, over [X] years, a significant reputation in the relevant location. Tell them that you have become aware of their business and that, in view of your reputation and goodwill, that the use of their business name is XXXXX XXXXX cause confusion in the minds of the public and your customers, along with damage to your business.
You can then ask them to amend their name. If you want to take a robust approach in your correspondence, you can also advise them that if the matter is not addressed you may need to give consideration to taking proceedings for an injunction and damages for lost profit, costs and interest.
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