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Search has played a fundamental role in computer game research since the very beginning. And while online search has been commonly used in perfect information games such as Chess and Go, online search methods for imperfect information games have only been introduced relatively recently. This paper addresses the question of what is a sound online algorithm in an imperfect information setting of two-player zero-sum games. We argue that the~fixed-strategy~definitions of exploitability and $ε$-Nash equilibria are ill-suited to measure an online algorithm's worst-case performance. We thus formalize $ε$-soundness, a concept that connects the worst-case performance of an online algorithm to the performance of an $ε$-Nash equilibrium. As $ε$-soundness can be difficult to compute in general, we introduce a consistency framework -- a hierarchy that connects an online algorithm's behavior to a Nash equilibrium. These multiple levels of consistency describe in what sense an online algorithm plays "just like a fixed Nash equilibrium". These notions further illustrate the difference between perfect and imperfect information settings, as the same consistency guarantees have different worst-case online performance in perfect and imperfect information games. The definitions of soundness and the consistency hierarchy finally provide appropriate tools to analyze online algorithms in repeated imperfect information games. We thus inspect some of the previous online algorithms in a new light, bringing new insights into their worst-case performance guarantees.