Is it possible for either the company’s registered address or company’s correspondence address in the API to be an overseas address?
For “standard” UK companies there must be a UK address registered - see the rules here:
However the Companies House dataset actually includes a wider range of entities and so you will indeed find some overseas addresses. The following cases are relevant - I can’t guarantee they are exhaustive though (you can explore these easily by using Advanced Companies Search and changing the company type):
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Foreign companies. These generally have more than one entry in Companies House - the foreign company itself (company numbers start “FC”) and “UK establishments” (can be more than one I believe) (with company numbers starting “BR”). Example: NORTH ROW INVESTMENTS LLC overview - Find and update company information - GOV.UK
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Overseas Entities - (information here) - these have company numbers starting “OE” example: FLEMING INDEPENDENT TRUSTEES LIMITED overview - Find and update company information - GOV.UK
You might expect that what used to be EEIG and SE companies (now renamed) would have e.g. European addresses but the ones I have seen all have UK addresses. However in a couple of cases someone has added a different country name (presumably where they relate to) giving an odd address which is clearly in the UK (including postcode) but with a different country (Spain in my example) e.g.: HCC SHARED SERVICES, A.E.I.E. overview - Find and update company information - GOV.UK
As well as exploring in advanced search you can see this in the Companies House bulk data.
(Note: “Correspondence address” is the same field as “Registered Office” and I believe that is only for Overseas Entities.)