We show that many tame modules of the quantum toroidal $\mathfrak{ gl}_2$ algebra can be explicitly constructed in a purely combinatorial way using the theory of $q$-characters. The examples include families of evaluation modules obtained from analytic continuation and automorphism twists of Verma modules of the quantum affine $\mathfrak{ gl}_2$ algebra. The combinatorial bases in the modules are labeled by colored plane partitions with various properties.
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