Load data is always listed for .223. No one makes 5.56mm dies that I have ever seen. The primary differences, as you may know, are in the chamber dimensions of the different rounds, and there MAY be differing chamber pressures in some rifles due to these dimension differences.
.223 Rem and 5.56x45 NATO are dimensionally identical, so common .223 loading dies can be used for either round. The difference is that the 5.56x45 NATO chamber has a longer leade.
Since the bullet has more free-travel through the leade in a 5.56x45 NATO chamber, the peak pressure is lower: the bullet already has a running start by the time it engages rifling.
In a typical .223 Rem chamber, the leade may be short enough to require excess pressure to get the bullet moving down the bore. This is a side effect of the common "match" chamber where the bullet is seated as close to the rifling as possible.
There are a few loading manuals that list both a "service rifle" .223 and 5.56x45 NATO. In these manuals the load data is identical except the powder charges are slightly higher in 5.56x45 NATO.
That's why it's safe to shoot .223 Rem in a gun chambered for 5.56x45 NATO but not vice-versa.