Trim the small white tendons from the ends of the tenderloins as needed and discard.
Combine the flour, 1 teaspoon salt and a few grinds of black pepper in a shallow dish (a pie plate works well for this). Dredge the tenderloins, shake off the excess flour and set aside.
Heat the butter in a large skillet over medium heat. Add the ginger and shallot and sauté until the ginger is fragrant and the shallots softened, 1 to 2 minutes.
Add the tenderloins and cook, turning several times, until barely golden, 1-1/2 to 2 minutes per side.
Add the pomegranate juice and continue cooking, turning the tenderloins several times, until the liquid thickens slightly and the chicken is cooked through, 2 to 3 minutes additional. Transfer the chicken to a serving platter.
Whisk the half-and-half into the pomegranate sauce and cook until the mixture is smooth, 1 to 2 minutes. Stir in the pomegranate arils, then spoon the sauce over the chicken.