If we have an EU client who is the Controller of data (let's assume client data about previous purchases for a retail/ecommerce store) and we must become a Processor of this data, when are we considered a Controller of the data? I'm assuming that we technically could not be the Controller since we do not have the relationship with the client directly, but I'm a little unsure of the exact nature of that relationship.
This depends on what activities you perform. If you process personal data on behalf of the Controller and if you process data based on the controller`s instructions, you are a data processor.
A controller is an entity who, alone or jointly with others, determines the purposes and means of the processing of personal data. In other words, the controller decides “what” personal data will be processed for and “how” it will be done.
A processor is an entity who processes personal data on behalf of a controller. An example might be a company that processes your payroll or a cloud provider that offers data storage. Ho wever, in more complex relationships it can be difficult in practice to work out if someone acts as controller or processor.