The paper studies multi-competitive continuous-time epidemic processes. We consider the setting where two viruses are simultaneously prevalent, and the spread occurs due to individual-to-individual interaction. In such a setting, an individual is either not affected by any of the viruses, or infected by one and exactly one of the two viruses. One of the equilibrium points is the coexistence equilibrium, i.e., multiple viruses simultaneously infect separate fractions of the population. We provide a sufficient condition for the existence of a coexistence equilibrium. We identify a condition such that for certain pairs of spread matrices either every coexistence equilibrium lies on a line that is locally exponentially attractive, or there is no coexistence equilibrium. We then provide a condition that, for certain pairs of spread matrices, rules out the possibility of the existence of a coexistence equilibrium, and, as a consequence, establishes global asymptotic convergence to the endemic equilibrium of the dominant virus. Finally, we provide a mitigation strategy that employs one virus to ensure that the other virus is eradicated. The theoretical results are illustrated using simulations.