There is an Asian supermarket in the middle of the Carmel market (shuk) called East-West.
They have soy sauce and other Asian sauces as well as galangal, ginger and fresh tumeric (last time I was there it was out of stock so best to call first) http://www.eastwest-food.co.il/ntext....
I don't know if this question is still relevant, but the best place for Chinese grocers is on or around Newe Sha'anan St. near the central bus station. The biggest one is a place called Dragon on Rosh Pina St. just off Newe Sha'anan. This is the only place in Tel Aviv where I was able to find peanut oil.
There are also two good Indian grocers on Levinski St. at or next to the central bus station.
There are 2 Asians store in the shuk (one in the centre, behind the stalls and one at the far end across from the park). There are also some Philipino stores in the New Central Bus station and an Asian spice store in the Levinsky Street Market. What are you looking for?