A wrappable textile sleeve for protecting an elongate member and method of construction thereof is provided. The sleeve includes a plurality of warp filaments and at least one weft filament woven with one another to form a woven wall. The woven wall has opposite sides extending lengthwise between opposite ends. The opposite sides are wrappable about a central longitudinal axis into overlapping relation with one another to bound the elongate member within a cavity of the sleeve. At least one warp and/or weft filament is provided as a bulky, fluffy multifilament, wherein the bulky, fluffy multifilament is woven to form at least one float extending over 2 or more weft and/or warp yarns, wherein the at least one float provides loops sufficient to readily fix themselves to hooks of a standard hook and loop fastener.