If your felloship is over, but your Ph.D. is not, you can ask for unemployment benefit in Belgium. Your are not required to be Belgian for this. It is enough that you worked XXX days in the last years in Belgium. Another important requirement is that you payed the relative taxes within your fellowship, and this is usually the case. Probably you did not notice it because it was done directly by ULB.
There are several entities involved in the unemployment benefit. The first, and probably the most important for you is the ONEM. They are the ones who actually have the money. You will seldom deal with them though.
The people that will directly give you the money will be either a trade union (CSC, FGTB or CGSLB) or the public agency, the CAPAC. The advantage of a trade union is that usually the inscription process is faster, but you will probably have to pay a fee each month. The CAPAC is free of charges, but waiting in their offices can be really painful.
Finally, there is the ORBEM. This is just an organisation that searches for jobs for you. You MUST inscribe to it to obtain the unemployment benefit. It is not clear if you have to inscribe within the 8 days after your last pay or after your inscription to, say, CAPAC. I suggest to do it immediately, and then inscribe to the CAPAC or to the unions. The inscription can be done on their web site.