Given $k$ iid $n\times n$ random Haar unitaries $U_1,\ldots , U_k$ and $l$ a fixed integer, we consider the joint behavior of $U_1^{\otimes l},\ldots , U_k^{\otimes l}$ and show that this sequence of $k$-tuples is almost surely strongly asymptotically free in the large $n$ limit. Strong asymptotic freeness is a particular case of strong convergence, which ensures the absence of outliers for a matrix model obtained from a non-commutative polynomial in the $k$-tuple  We will explain our motivations, describe some variants of our result and some applications in asymptotic representation theory. We will also elaborate on a few salient aspects of the proof, such as a theory of matrix valued non-backtracking operators, and a new inequality between moments of gaussian and unitary matrices. This talk is based on joint work in preparation with Charles Bordenave.