First of all, this is a local law question. To illustrate why, let's look at two different places, namely the United States and Indonesia. The US follows common law in the Anglo-French tradition, except Louisiana which adopted civil law and the Napoleonic Code. Marriage is a state law issue in the US and there is little Federal involvement in that issue.
In Indonesia, marriage is primarily a religious issue, and the federal government basically rubber stamps religious determinations on who is married, but civil law actions are required for divorce (and Indonesia follows, roughly Dutch/Roman civil law in this regard, with some influence from India). Being married or being separated in both cases is very different thing, because there are questions of whether joint assets exist or how they are divided, and the two countries may have different opinions on that.
The processes for handling these questions are different in both countries (and even between some states in the US!). So you want to check with local lawyers in both places. Before you do, however, make sure you are prepared to talk about the following things:
What you mean by living separately. Is this a legal arrangement regarding assets, possibly preceding a divorce? Or is it a practical temporary arrangement preceding moving back together? This may have profound implications regarding needing to register, tax status, etc.
If it is a legal arrangement, what are the details that have been agreed upon? Are there legal implications there?
These thins depend on specifics and local laws, and so local lawyers are the ones to ask.