Summary: Recently it was shown how necessary and sufficient information for solving an orthogonally invariant linear approximation problem \(AX\approx B\) with multiple right-hand sides can be revealed through the so-called core problem reduction; see [\textit{I. Hnětynková} et al., SIAM J. Matrix Anal. Appl. 34, No. 3, 917--931 (2013; Zbl 1279.65041)]. The total least squares (TLS) serves as an important example of such approximation problem. Solvability of TLS was discussed in the full generality in [\textit{I. Hnětynková} et al., SIAM J. Matrix Anal. Appl. 32, No. 3, 748--770 (2011; Zbl 1235.15016)]. This theoretical study investigates solvability of core problems with multiple right-hand sides in the TLS sense. It is shown that, contrary to the single right-hand side case, a core problem with multiple right-hand sides may not have a TLS solution. Further possible internal structure of core problems is studied. Outputs of the classical TLS algorithm for the original problem \(AX\approx B\) and for the core problem within \(AX\approx B\) are compared.