A process for refurbishing a worn surface of a shroud support component of a turbomachine, as well as a shroud support component refurbished with the process. The process generally entails removing a surface region of the worn surface so as to define a repair surface on the component. A braze tape formed from a slurry to comprise a braze material and a wear-resistant alloy is then applied to the repair surface, followed by a heat treatment to cause the braze tape to diffusion bond to the repair surface so as to define a built-up surface. The built-up surface is then machined to define a wear-resistant coating on the component.