A system is provided that includes a compressor section, a turbine section, a first engine and a second engine. The compressor section includes a compressor rotor. The turbine section includes a turbine rotor configured to drive rotation of the compressor rotor. The first engine includes a first engine inlet, a first engine outlet and a first engine combustion zone fluidly coupled with and between the first engine inlet and the first engine outlet. The first engine inlet is fluidly coupled with and downstream of the compressor section. The first engine outlet is fluidly coupled with and upstream of the turbine section. The second engine includes a second engine inlet, a second engine outlet and a second engine combustion zone fluidly coupled with and between the second engine inlet and the second engine outlet. The second engine inlet is fluidly coupled with and downstream of the compressor section.