the tamarind I looked up is Tamarindus indica. it is classified as Diffuse-porous; medium to large pores in no specific arrangement.
chasing a ring is recommended on trees that are classified as ring porous, where there is a distinct layer of porous wood that does not do well as the back of the bow. I would try it unbacked and see, especially if your piece is as described above. If it is what you often have to work with, you will never know unless you try to tiller it unbacked . often, Diffuse-porous woods are very difficult to chase a ring on because they do not have much of a "ring" to chase.
Of course, if your goal is to make this stave into a bow and be done with tamarind, then the extra nsurance of a backing is OK