So I just tested your steps and did not get the same results you are reporting. I did test on a newer device (iPad Air), but I doubt that is the issue since you also mention turning the texture into a sprite the tilemap renders correctly.
What with that said though, the default material used for tilemaps is a Translucent Unlit Sprite Material. The iPhone 5s should display this correctly, but I am not 100% sure if that is the case. Do you have another iOS device you can test the issue?
Cheers,