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In the present paper we consider multi-scalar extension of Einstein-Gauss-Bonnet gravity. We focus on multi-scalar Einstein-Gauss-Bonnet models whose target space is a three-dimensional maximally symmetric space, namely either $\mathbb{S}^3$, $\mathbb{H}^3$ or $\mathbb{R}^3$, and in the case when the map $\text{\it spacetime} \to \text{\it target space}$ is nontrivial. We prove numerically the existence of black holes in this class of models for several Gauss-Bonnet coupling functions, including the case of scalarization. We also perform systematic study of a variety of black hole characteristics and the space-time around them, such as the area of the horizon, the entropy and the radius of the photon sphere. One of the most important properties of the obtained solutions is that the scalar charge is zero and thus the scalar dipole radiation is suppressed which leads to much weaker observational constraints compared to the majority of modified theories possessing a scalar degree of freedom. For one of the coupling functions we could find branches of scalarized black holes which have a nontrivial structure -- there is non-uniqueness of the scalarized solutions belonging to a single branch and there is a region of the parameter space where most probably stable scalarized black holes coexist with the stable Schwarzschild black holes. Such a phenomena can have a clear observational signature.