A recent dividend on NYSE:SNE (Sony) is described as "Will not be quoted ex". What does this actually mean? Does the dividend have an ex-date or not? Which day would be the last day to buy this stock to obtain the dividend?
The ex indicator is meant to be a help for market participants. On the ex-day orders will go into a different order book, the ex order book, which at the start of the ex day will be totally empty, i.e. no orders from the non-ex day book have been copied over.
Why does this help? Well imagine you had a long-standing buy order in the book, well below the current price, and now the share price halves due to a 2-for-1 split, would you want to see your order executed?
If so, your order should have gone into the ex-book which is only active on the ex-day (and orders in the ex book are usually copied over to the normal book on the day after the ex-day but this is exchange-specific).
Think of it as an additional safety net to tell the exchange: "I know what I'm doing: I want to buy this stock totally overpriced after the 2-for-1 split".
Now some exchanges and/or some securities (mostly derivatives) linked with the security in question don't have this notion of ex or the ex-book, and they will tell you by "will not be quoted ex" or "the ex indicator is missing".
In your case (SNE) it is a sponsored ADR, the ex-date was Mar 28 2016, one day before the ex date of the Japanese original. According to my understanding of NYSE rules, there is no specific rule for or against omitting the ex-indicator. It seems to be a decision on a case by case basis. Looking through the dividends of other Japanese ADRs I drew the conclusion none of them have an ex-book and so all of them are announced as: "Will not be quoted ex by the exchange". Again, this is based on my observations.
One occastion where "will not be quoted ex" is used is when a corporate action is occurring such as a spin-off. In such a case, the rights to, and the spin-off itself may be quoted separately on the home country exchange. However, if the company is based abroad, it may not be worth the expense for them to have an additional securities listing on the local (US) exchange.
For example: In November 2016, Yamana Gold (TSX: YRI, NYSE: AUY) announced it will have an initial public offering of a spin-off (Brio Gold, to be listed on TSX as BRIO). Existing shareholders received a right to one share of the spin-off for every 16 shares they held of YRI (or AUY). These rights were separately traded in advance of the IPO of the spin-off on TSX under "YRI.RT", but the prospectus they stated that the rights "will not be quoted ex" on NYSE, i.e. there was no separate listing on NYSE for these rights.
The wording seems counter-intuitive, but I suspect that is the attorneys who were preparing the prospectus used those specific words as they may have a very specific meaning (e.g. from a statute or previous case).