We study the four plactic-like monoids that arise by taking the meets and joins of stalactic and taiga congruences. We obtain the combinatorial objects associated with the meet monoids, establishing Robinson-Schensted-like correspondences and giving extraction and iterative insertion algorithms for these objects. We then obtain results on the sizes of classes of words equal in plactic-like monoids, show that some of these monoids are syntactic, and characterise their equational theories.