Tilted-axis rotation, arising from Fermi-aligned configurations, has been observed for the first time to cause backbending in an odd-proton nucleus. In ¹⁸¹Re, two t-bands are found to be energetically favored relative to the usual rotation-aligned s-bands, presenting an alternative form of cold nuclear rotation. Interactions between the bands are weak, and unambiguous comparisons with tilted-axis-cranking calculations can be made.