Hi rg2013a,
The first place to start when you are looking for a new mattress is post #1 here. It will give you the information, steps, and guidelines that you will need to make the best possible decisions. It will also help you know why to avoid brand shopping and focus on materials instead.
Post #2 here has more about trying to 'match" another mattress but it can be very difficult because feel is so subjective but in the case of the Tempurpedic lineup … there are several companies that have put some care into approximating either the quality or materials, the mattress design, or the feel of their mattresses (all of these are not the same thing) and in some cases have approximated all three.
The Rhapsody is a memory foam mattress which is very different from a latex mattress. They are apples and oranges in terms of how they feel and perform and latex won’t ever feel like memory foam (or vice versa).
Both Brooklyn Bedding and Rocky Mountain Mattress are members of this site which means that like all the members here I believe they are among the best quality and value in the country.
When you get to step #3 … if you let me know your city or zip I’d be happy to let you know about some of the better options or possibilities I’m aware of in your area.
Phoenix