An all-fiber, dual Mach-Zehnder interferometer (MZI) based on an acoustically modulated hollow core fiber (HCF) is experimentally demonstrated for the first time. By attaching an acoustic driver between the fixed ends of an HCF, we fabricated two acousto-optic modulators (AOMs) with distinct driver positions, allowing synchronization of two in-line MZIs inside the HCF. The first MZI is set by two acoustic long-period gratings separated by a second MZI formed at the fiber and driver-attaching region. We show that this setup enables frequency-tuning of the coupling between the fundamental and higher-order modes in the HCF. Additionally, we simulate and analyze the HCF modal couplings and MZIs' modulated spectra under distinct device parameters using the transfer matrix method. The new AOM-MZI enables tuning of the MZI's free spectral range by adjusting 1 Hz of the electrical frequency, which is promising to modulate multiwavelength filters, sensors, and fiber lasers.