Once you upgrade to the Rarefied Attunement Structure, you can begin tuning up Monsters into their Rare forms. To start, select the Structure and then Tune Up to add an eligible Monster. This is also accessible from the center of the Island carousel in the standard Attunement menu.

While attuning a Meeb that matches one of the selected Monster's Current Tunable Elements, there's a chance to tune up the Monster's matching Element at the same time (for an additional Shard cost). You can only tune up Elements that are currently available for tuning up.

If you select your Rarefied Attunement Structure, select Tune Up and look in the bottom right of the menu, you will see Current Tunable Elements with the available Elements beside it. For example, if Rare Blarret is available in the Market, the Plasma, Shadow and Poison Elements will be available for tuning up. If you select a Monster which has any of the Current Tunable Elements, you can start attuning Meebs that match one of those Elements to attempt tuning them up. This costs 10,000 additional Shards to attempt. 

If you successfully tune up an Element, you will see a pop-up appear after the Meeb completes the Attunement process which indicates that Element has been tuned up. You can confirm which Elements you have successfully tuned up by selecting your Rarefied Attunement Structure and looking at the Elements beside a Monster’s name in the list. Any Elements that have been successfully tuned up will have a blue/green glowing outline.

When all of the Monster's Elements have been tuned up, it will transform into a Rare!