Multi-product exporters choose their product mix focusing on their best-performing products. Although their product mix varies across countries (the fickle fringe), the interdependence in demand or production technology making vectors of products systematically co-exported leads to commonalities in this mix across destinations (the stable core). In order to uncover the determinants of the fickle and stable parts of firm export product mix, we use a cross section of firm-product-destination level French and Italian data, taking explicitly into account the choice of not exporting a product to a destination. Using dissimilarity measures instead of rank correlations, we observe a great deal of variability among the product-mixes a firm exports to different destinations. We show that market size, but also the market positioning of a firm and market structure explain part of this observed variability. At the same time, together with this fickle part, we highlight the existence of a stable component among the diverse product-mixes exported. The probability of exporting this core set of products increases with the size of the destination market and with the ability to match demand, but is inversely related to market concentration.