Latent Markov models (LMMs) are commonly used to analyze longitudinal data from multiple diagnostic tests. LMMs consist of a structural model for the latent infection state, defining probabilities for the initial state and transmission between states, and a measurement model for the observed test results, defining the item response probabilities and thus test sensitivities and specificities. LMMs typically assume that tests are independent conditional on the latent infection state. This is likely to be violated for tests using similar technologies. We introduce random effects to relax the conditional independence assumption and we derive a generalization of the basic LMM for an application to Salmonella infection data. We analyze longitudinal data from four molecular PCR tests and a stool culture test from patients in Blantyre, Malawi. To assess the tests' performances, we consider basic and mixed LMMs, both with time homogeneous and heterogeneous transition probabilities. We compare the different models and discuss technical considerations. A PCR assay using primers from the TTR gene achieves the best sensitivity / specificity trade-off.