High quality heteroepitaxy of CdTe is challenging due to lattice mismatches of CdTe with many substrates. Herein, we demonstrate the epitaxial growth of single crystalline CdTe films, in multilevel island format, on mica using metalorganic chemical vapor deposition, regardless of large in-plane lattice mismatch (\ensuremath{\sim}13%) between CdTe(111) and mica(001). X-ray and electron diffractions suggest that CdTe is epitaxially aligned with mica: out-of-plane $\mathrm{CdTe}(111)\text{//mica}(001)$ and in-plane CdTe $[\overline{1}2\overline{1}]$//mica $[010]$. Full-width-at-half-maximum (FWHM) of x-ray rocking curve and FWHM of x-ray azimuthal in-plane angular dispersion of CdTe are shown to be $0.{11}^{\ensuremath{\circ}}$ and $0.{38}^{\ensuremath{\circ}}$, respectively, better than most CdTe films reported. Electron backscattering diffraction shows that CdTe grains are tens of \ensuremath{\mu}m and, if twin boundaries are excluded, in excess of 250 \ensuremath{\mu}m in size. In contrast to the belief that overlayer growth on mica is purely through van der Waals interaction, our first-principle calculations uncover that van der Waals interaction only contributes to 20% of the total interfacial energy, and 80% of the interfacial energy comes from chemical interaction. We believe such a strong chemical interaction is accountable for the high-quality epitaxy. The demonstration of epitaxial growth of high-quality semiconductor on mica with a large lattice mismatch creates opportunities for flexible optoelectronic devices.