Rebuilding Economies and Creating Opportunities for More People To Excel Act or the RECOMPETE Act This bill establishes a grant program to assist economically distressed local communities and labor markets that meet specified economic criteria, including employment criteria. The Economic Development Administration (EDA) must award 10-year grants to a local government, tribal government, economic development district, or consortium of local government units located in such areas. Each grant recipient must develop and implement a comprehensive strategy to address the economic challenges specific to the area. The grants may be used to increase employment opportunities, increase local per capita income, support economic development, and develop infrastructure and housing. Additionally, the EDA must periodically evaluate each grant recipient based on certain benchmark criteria and annually report on the implementation of the program. The bill specifies formulas for determining the grant amount for each community or market and the federal share of the cost of each program or activity conducted under the program.