SAP BP: R1 473 during Business Partner create/change

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Symptom
When you try to create or change a Business Partner through web UI or through Business Partner BAPIs, you encounter an error R1 473. This happens only when you have extended the domain BU_CONTACT.
Other terms
BU_CONTACT, BAPI_BUPA_CREATE_FROM_DATA, BUPA_CREATE_FROM_DATA, BAPI_BUPA_CENTRAL_CHANGE, BUPA_CENTRAL_CHANGE, BUP_CHECK_CONTACT, R1 473, Business partner &1:Invalid value &2 for field &3, BAPIBUS1006_CENTRAL-CONTACTALLOWANCE, [...]