Markov chain Monte Carlo (MCMC) methods, specifically samplers based on random walks,often have difficulty handling target distributions with complex geometry such as multi-modality.We propose an adaptive multiple-try Metropolis algorithm designed to tackle such problems bycombining the flexibility of multiple-proposal samplers with the user-friendliness and optimalityof adaptive algorithms. We prove the ergodicity of the resulting Markov chain with respectto the target distribution using common techniques in the adaptive MCMC literature. In aBayesian model for loss of heterozygosity in cancer cells, we find that our method outperformstraditional adaptive samplers, non-adaptive multiple-try Metropolis samplers, and various moresophisticated competing methods.