We consider the one-dimensional totally asymmetric simple exclusion process with initial product distribution with densities 0=<ρ 0 <ρ 1 <...<ρ n =<1 in (-~,c 1 - 1 ), [c 1 - 1 ,c 2 - 1 ),...,[c n - 1 ,+~), respectively. The initial distribution has shocks (discontinuities) at - 1 c k , k=1,...,n, and we assume that in the corresponding macroscopic Burgers equation the n shocks meet in r * at time t * . The microscopic position of the shocks is represented by second class particles whose distribution in the scale - 1 / 2 is shown to converge to a function of n independent Gaussian random variables representing the fluctuations of these particles ''just before the meeting''. We show that the density field at time - 1 t * , in the scale - 1 / 2 and as seen from - 1 r * converges weakly to a random measure with piecewise constant density as ->0; the points of discontinuity depend on these limiting Gaussian variables. As a corollary we show that, as ->0, the distribution of the process at site - 1 r * + - 1 / 2 a at time - 1 t * tends to a non-trivial convex combination of the product measures with densities ρ k , the weights of the combination being explicitly computable.