Hi MikeL60,
The mattress shopping tutorial (which is where I would start your research) includes this link to a list of some of the better online memory foam providers that I’m aware of. Most of them are very familiar with the various Tempurpedic mattresses and would be able to tell you which of their mattresses are closest to a particular model in the Tempurpedic lineup and some of them sell a mattress that is specifically designed and listed on their sites as being closely comparable to the Rhapsody Luxe or the Rhapsody Breeze. Post #9 here and the posts it links to also has more about the different ways that one mattress can “match” another one which you can use as a reference in your conversations with them.
Latex is a completely different material than memory foam and they are not really comparable to each other in terms of “feel” so you won’t find a latex mattress that has a similar “feel” to a memory foam mattress. There is more about how latex compares to memory foam in post #2 here although both types of mattresses can be used in a mattress that is a good match for a particular person in terms of PPP (Posture and alignment, Pressure relief, and Personal preferences). If you are considering latex (which is certainly worthwhile to include in your research) I would test a few latex mattresses locally so you have a better sense of how the two materials compare at least in general terms (both mattress categories have a wide range of different designs and firmness levels so one mattress won’t represent the entire category). Since the choice between latex and memory foam is a preference choice … your own personal experience is the best way to know which one you tend to prefer.
If you let me know your city or zip code I’d be happy to let you know of the better options or possibilities I’m aware of in your area so you can do some personal testing on latex mattresses (assuming there are some available that are within reasonable driving distance).
Phoenix