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Hard X-ray observations with the Neil Gehrels Swift Observatory Burst Alert Telescope (BAT) reveal superorbital modulation in the wind-accreting supergiant high-mass X-ray binary (HMXB) 4U 1538-52 at a period of 14.9130 +/- 0.0026 days that is consistent with four times the 3.73 day orbital period. These periods agree with a previously suggested correlation between superorbital and orbital periods in similar HMXBs. During the ~14 years of observations the superorbital modulation changes amplitude, and since ~MJD 57,650 it was no longer detected in the power spectrum, although a peak near the second harmonic of this was present for some time. Measurements of the spin period of the neutron star in the system with the Fermi Gamma-ray Burst Monitor show a long-term spin-down trend which halted towards the end of the light curve, suggesting a connection between dP(spin)/dt and superorbital modulation, as proposed for 2S 0114+650. However, an earlier torque reversal from INTEGRAL observations was not associated with superorbital modulation changes. B and V band photometry from the Las Cumbres Observatory reveals orbital ellipsoidal photometric variability, but no superorbital optical modulation. However the photometry was obtained when the 14.9130 day period was no longer detected in the BAT power spectrum. We revisit possible superorbital modulation in BAT observations of IGR J16393-4643 but cannot conclusively determine whether this is present, although is not persistent. We consider superorbital modulation mechanisms, and suggest that the Corotating Interaction Region model, with small deviations from orbital synchronization, appears promising.