I don't know what the engraving on your watch is like, but the 113.603 was issued to the Israeli Defence forces, I believe the naval units (Israel does have a small, but historically successful navy).
The Israeli 113.603 have very distinctive engraving with thick letters (often starting with "M") engraved, usually, in an italic font. I have seen IDF Autavia's with plain back and ones with the engraved Heuer shield on the back of the later editions.

I don't have any definitive information like unit numbers or personal accounts, but the IDF is usually pretty reticent about such details so it is not surprising. I bought my one from a Israeli dealer who specialises in ex-military equipment (not especially watches). He was selling two of them, I think Alistair bought the other one, he might have more details.

I'm having hassles with my homepages unit, so can't post a picture right now. Actually, if anyone can tell me how to link to a photo in apple's me.com galleries, it would be helpful. I'm sure Chuck would have been able to help me!.

regards
Shaun