A formula is given for the complex intersection number of real cycles on a singular real algebraic variety \(X\) whose singularities are topologically rational, in terms of the local intersection numbers and the compactly supported Euler characteristics of the strata of a semi-algebraic Whitney stratification of \(X\). The author is motivated by an extension of the Arnold inequalities to singular varieties. The applicability of the formula is nicely illustrated for \(X\) a double cover of the projective space branched along a generic hypersurface.