The solution of the perturbed Tanaka-equation is pathwise unique

The Tanaka equation $d{X}_t=\mathrm{sign}(X_t)d{B}_t$ is an example of a stochastic differential equation (SDE) without strong solution. Hence pathwise uniqueness does not hold for this equation. In this note we prove that if we modify the right-hand side of the equation, roughly speaking, with a strong enough additive noise, independent of the Brownian motion B, then the solution of the obtained equation is pathwise unique.