The unit system that is coherent with both inches and pounds is the Imperial system, which is commonly used in the United States and a few other countries. In this system, the unit of force is the pound-force (lbf).
The pound-force (lbf) is a unit of force defined as the force required to accelerate a mass of one pound by one foot per second squared (1 lb·ft/s²). It is derived from the pound-mass (lbm) and the acceleration due to gravity (32.17405 ft/s²).
So, in the coherent Imperial system, the unit of force is the pound-force (lbf), which is used alongside inches for distance measurements and pounds for mass measurements.