The tangential end fibration of an aspherical Poincaré complex

We construct a sphere fibration over a finite aspherical Poincaré complex $X$, which we call the tangential end fibration, under the condition that the universal cover of $X$ is forward tame and simply connected at infinity. We show that it is tangent to $X$ if the formal dimension of $X$ is even or, when the formal dimension is odd, if the diagonal $X\to X\times X$ admits a Poincaré embedding structure.