We use a deterministic model to study two competing viruses spreading over a two-layer network in the Susceptible–Infected–Susceptible (SIS) framework, and address a central problem of identifying the winning virus in a “survival-of-the-fittest” battle. Almost all existing conditions ensure that the same virus wins regardless of initial states. In the present paper, we ask the following question: can we systematically construct SIS bivirus networks with an arbitrary but finite number of nodes such that either of the viruses can win the survival-of-the-fittest battle, depending on the initial states? We answer this question in the affirmative. More specifically, we show that given almost any network layer of one virus, we can (using our proposed systematic four-step procedure) construct the network layer for the other virus such that in the resulting bivirus network, either of the two viruses can win the survival-of-the-fittest battle. Conclusions from numerical case studies, including a real-world mobility network that captures the commuting patterns for people between 107 provinces in Italy, illustrate and extend the theoretical result and its consequences.