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 of one of that Monster's currently Tunable Elements, there's a chance to also Tune Up the Monster's matching Element at the same time (for an additional Shard cost).

When all of the Monster's Elements have been Tuned Up, it will evolve into a Rare!