Light-matter interaction based on strong laser fields enables probing the structure and dynamics of atomic and molecular systems with unprecedented resolutions, through high-order harmonic spectroscopy, laser-induced electron diffraction, and holography. All strong-field processes rely on a primary ionization mechanism where electrons tunnel through the target potential barrier lowered by the laser field. Tunnel ionization is, thus, of paramount importance in strong-field physics and attoscience. However, the tunneling dynamics and properties of the outgoing electronic wave packets often remain hidden beneath the influence of the subsequent scattering of the released electron onto the ionic potential. Here, we present a joint experimental-theoretical endeavor to characterize the influence of sub-barrier dynamics on the amplitude and phase of the wave packets emerging from the tunnel. We use chiral molecules, whose photoionization by circularly polarized light produces forward-backward asymmetric electron distributions with respect to the light propagation direction. These asymmetric patterns provide a background-free signature of the chiral potential in the ionization process. We first implement the attoclock technique, using bicircular two-color fields. We find that, in the tunnel-ionization process, molecular chirality induces a strong forward-backward asymmetry in the electron yield, while the subsequent scattering of the freed electron onto the chiral potential leads to an asymmetric angular streaking of the electron momentum distribution. In order to access the phase of the tunneling wave packets, we introduce subcycle gated chiral interferometry. We employ an orthogonally polarized two-color laser field whose optical chirality is manipulated on a sub-laser-cycle timescale. Numerical simulations are used to interpret the electron interference patterns inherent to this interaction scheme. They show that the combined action of the chiral potential and rotating laser field not only imprints asymmetric ionization amplitudes during the tunneling process, but also induces a forward-backward asymmetric phase profile onto the outgoing electron wave packets. Chiral light-matter interaction thus induces subtle angular-dependent shaping of both the amplitude and the phase of tunneling wave packets.