For protein mutagenesis, a common expectation is that important positions will behave like on/off "toggle" switches (i.e., a few substitutions act like wildtype, most abolish function). However, there exists another class of important positions that manifests a wide range of functional outcomes upon substitution: "rheostat" positions. Previously, we evaluated rheostat positions located near the allosteric binding sites for inhibitor alanine (Ala) and activator fructose-1,6-bisphosphate (Fru-1,6-BP) in human liver pyruvate kinase. When substituted with multiple amino acids, many positions demonstrated moderate rheostatic effects on allosteric coupling between effector binding and phosphoenolpyruvate (PEP) binding in the active site. Nonetheless, the combined outcomes of all positions sampled the full range of possible allosteric coupling (full tunability). However, that study only evaluated allosteric tunability of "local" positions, i.e., positions were located near the binding sites of the allosteric ligand being assessed. Here, we evaluated tunability of allosteric coupling when mutated sites were distant from the allosterically-coupled binding sites. Positions near the Ala binding site had rheostatic outcomes on allosteric coupling between Fru-1,6-BP and PEP binding. In contrast, positions in the Fru-1,6-BP site exhibited modest effects on coupling between Ala and PEP binding. Analyzed in aggregate, both PEP/Ala and PEP/Fru-1,6-BP coupling were again fully tunable by amino acid substitutions at this limited set of distant positions. Furthermore, some positions exhibited rheostatic control over multiple parameters and others exhibited rheostatic effects on one parameter and toggle control over a second. These findings highlight challenges in efforts to both predict/interpret mutational outcomes and engineer functions into proteins.