It is well known that braid groups act naturally on (powers of) objects of a braided monoidal category. Here we describe a braided monoidal category giving rise to braid group representations by symplectic matrices studied in [J. Assion, Math. Z. 163 (3) (1978) 291-302; B. Wajnryb, Israel J. Math. 76 (3) (1991) 265-288]. In contrast to the "standard" examples of braided monoidal categories, such as categories of representations of quantum groups, the monoidal structure in our example is given by sum of vector spaces rather than tensor product. The braiding is given by a simple formula which allows easy generalizations leading to new finite quotients of braid groups.