There is no general statutory requirement to appoint a process agent. The requirement is contractual: if you are an overseas party to an English law agreement, the other side will almost always insist on one, because the Civil Procedure Rules recognise service by a contractually agreed method and that certainty is what they are buying.
Where the requirement really comes from
The Civil Procedure Rules allow a claim form to be served by a method or at a place the parties have agreed in their contract. A process agent clause is exactly that agreement: it fixes a place in England and Wales where service will be valid. So the driver is not a statute compelling you to appoint, it is the counterparty's entirely rational refusal to sign without a reliable domestic service route, backed by rules that make the arrangement effective.
When you will find it unavoidable in practice
Cross border lending, ISDA documentation, English law leases and property purchases by overseas buyers, guarantees supporting UK obligations and many commercial contracts all carry the clause as standard, frequently as a condition precedent. Some regulated and market frameworks also expect a UK service address in specific contexts. For practical purposes, if your agreement is governed by English law and you have no establishment here, treat the appointment as part of the cost of the deal, from £125 per year.
Litigation without an agent
Where no agent exists, a claimant must usually serve out of the jurisdiction, often under the Hague Service Convention, which can take months and adds cost and uncertainty for everyone, including you as defendant, since disputes drag and default risks rise. Our answer on what happens without an agent covers this in detail. This page is general information, not legal advice on your agreement.