Early diagnosis of breast tumors can in principle be achieved by jointly running electromagnetic and acoustic probings, more precisely Microwave (MW) and ultrasound (US). Indeed, such modalities are non-invasive, non-ionizing, low-cost, and proceed without registration for free pending breasts, not like other joint modalities that impose compression. Because of the strongly-contrasted electromagnetic parameters of breast constituents, MW yields high-contrast images of low resolution, and the converse with the US since faced with weakly refracting elements. The key benefit is the common breast structure, and fusion should produce images with both high contrast and resolution. A Bayesian formalism is chosen, and an unsupervised Joint Variational Bayesian Approximation or JVBA is developed. In it, edges hidden variables and hyper-parameters are automatically tuned along with the optimization. The mathematics is detailed in a general setting of fusion, then one proposes imaging of realistic MRI-derived breast slices. A wealth of numerical simulations from noisy single-frequency MW and multiple-frequency US data preserved from inverse crime yields means (i.e., breast maps) and variances of the unknown electromagnetic and acoustic parameter distributions as probabilistic realizations, evolutions of hyper-parameters, as well as global indicators of accuracy. Comparisons with a joint edge-preserving contrast source inversion (JCSI-EP) developed earlier in a deterministic framework illustrate the methodology.