Hi Chako,
I really don't know anything definitive about your Gretsch Synchromatic but I have made a few obsevations that may prompt a Gretsch expert to respond and clear things up a bit.
I have consulted both Jay Scott's and Tony Bacon's Gretsch Books as well as the catalogs in my collection and I can't find a picture of a guitar that matches yours. Both the Gretsch books focus on electrics although Jay does include a few early acoustic Synchromatics in his book.
My hunch is that your Synchromatic was originally acoustic.
The two main features of your guitar that are unusual compared to other Synchromatics I've seen are the headstock and the tailpiece.
The only headstock I have seen that resembles yours is on a 1949 Gretsch New Yorker. Here's a picture:
 
Based on the similarity of headstock, my vote right now is that your Synchromatic is also a 1949 model although I haven't seen a picture of a 1949 Synchromatic to confirm my hunch.
Now to the tailpiece. Every early Synchromatic tailpiece I have seen has a hinge like this one and has Gretsch engraved on the badge at the bottom. From what I can tell, your's is somewhat similar but with no hinge, badge or engraving. This may be a characteristic of the 1949 model, but I can't say since I've never seen one.
 
So, I'm still not absolutely sure what you have but maybe my post will prompt someone else to chime in.   
