Only two other tiles with this design are known. One is in the Isabella Stewart Gardner Museum (http://www.gardnermuseum.org/collection/tile_c7w3.asp) and the other is said to be in the Art Club, Providence, Rhode Island.
Although it has no exact parallels, the ogival lattice of this tile panel is similar in visual impact to a slightly earlier one on the interior of the Mosque of Rüstem Pasha, circa 1560 (Walter Denny, Gardens of Paradise. 16th Century Turkish Ceramic Tile Decoration, Turkey, 1998, pl. 10, pp. 36-37). The oval medallions, each framing a single bold red tulip create a similar powerful effect in each. The main difference comes that the medallions in the Rüstem Pasha example are framed with cloud bands whilst those in the present are alternated with circular medallions and undulating vine.
Ottoman art is no stranger to the ogival lattice. It is perhaps most commonly encountered in kemha weaving. Indeed in that medium, similar lattices with large tulip decoration are found. See for example, Nurhan Atasoy, Walter B. Denny, Louise W. Mackie and Hülya Tezcan, Ipek. The Crescent and the Rose. Imperial Ottoman Silks and Velvets, London, 2001, pp. 276-77. Other motifs from the present tile panel can also be echoed in Ottoman textiles. The fleshy serrated vine leaves with seven main points find a parallel in a velvet in the Freer Gallery of Art , dated circa 1600 (Yanni Petsopoulos (ed.), Tulips, Arabesques and Turbans, London, 1982, no. 132, p. 140). Similarly the circular medallions containing tulips and carnations issuing from a central rosette are found in a kemha of the second half of the 16th century in the Musée de la Mode et du Textile, Union Centrale des Art Décoratifs, Paris (inv. no. 18312, Atasoy et al, op. cit., 2001, p. 185).