In view of the recently seen dramatic effect of quenched random bonds on tricritical systems, we have conducted a renormalization-group study on the effect of quenched random magnetic fields on the tricritical phase diagram of the spin-1 Ising model in d = 3. We find that random fields convert first-order phase transitions into second-order phase transitions more effectively than random bonds. The coexistence region is extremely flat, attesting to an unusually small tricritical exponent beta(u). The random-field phase diagram is strikingly asymmetric. To accommodate this asymmetry, the second-order boundary exhibits reentrance. [S0031-9007(99)08751-7].