Summary: In this paper, a nonsmooth optimization method for locally Lipschitz functions on real algebraic varieties is developed. To this end, the set-valued map \(\varepsilon\)-conditional subdifferential \(x\to\partial_{\varepsilon}^Nf(x):=\partial_{\varepsilon}f(x)+N(x)\) is introduced, where \(\partial_{\varepsilon}f(x)\) is the Goldstein-\(\varepsilon\)-subdifferential and \(N(x)\) is a closed convex cone at \(x\). It is proved that negative of the shortest \(\varepsilon\)-conditional subgradient provides a descent direction in \(T(x)\), which denotes the polar of \(N(x)\). The \(\varepsilon\)-conditional subdifferential at an iterate \(x_{\ell}\) can be approximated by a convex hull of a finite set of projected gradients at sampling points in \(x_\ell+\varepsilon_{\ell} B_{T(x_{\ell})}(0,1)\) to \(T(x_{\ell})\), where \(T(x_{\ell})\) is a linear space in the Bouligand tangent cone and \(B_{T(x_{\ell})}(0,1)\) denotes the unit ball in \(T(x_{\ell})\). The negative of the shortest vector in this convex hull is shown to be a descent direction in the Bouligand tangent cone at \(x_{\ell}\). The proposed algorithm makes a step along this descent direction with a certain step-size rule, followed by a retraction to lift back to points on the algebraic variety \(\mathcal{M}\). The convergence of the resulting algorithm to a critical point is proved. For numerical illustration, the considered method is applied to some nonsmooth problems on varieties of low-rank matrices \(\mathcal{M}_{\leq r}\) of real \(M\times N\) matrices of rank at most \(r\), specifically robust low-rank matrix approximation and recovery in the presence of outliers.