The key characteristic that determines when consolidated financial statements should be prepared is:

The key characteristic that determines when consolidated financial statements should be prepared is Control.

Here are the steps to understand this:

1. Consolidated Financial Statements: These are financial statements that roll up multiple accounts of a company and its subsidiaries into one set of financial statements.

2. Control: In the context of consolidated financial statements, control refers to the power to govern the financial and operating policies of an entity so as to obtain benefits from its activities. This is typically evidenced by ownership of more than 50% of the voting rights of the entity.

3. When to Prepare Consolidated Financial Statements: A parent company should prepare consolidated financial statements when it has control over one or more other entities (subsidiaries). This is regardless of whether the control is exercised directly by the parent or indirectly through subsidiaries.

So, the key characteristic that determines when consolidated financial statements should be prepared is whether the parent company has control over the subsidiary entities.