The late-time light curves of Type Ia supernovae (SNe Ia), observed >900 days after explosion, present the possibility of a new diagnostic for SN Ia progenitor and explosion models. First, however, we must discover what physical process (or combination of processes) leads to the slow-down of the late-time light curve relative to a pure 56Co decay, as observed in SNe 2011fe, 2012cg, and 2014J. We present Hubble Space Telescope observations of SN 2015F, taken ~600-1040 days past maximum light. Unlike those of the three other SNe Ia, the light curve of SN 2015F remains consistent with being powered solely by the radioactive decay of 56Co. We fit the light curves of these four SNe Ia in a consistent manner and measure possible correlations between the light curve stretch - a proxy for the intrinsic luminosity of the SN - and the parameters of the physical model used in the fit (e.g., the mass ratio of 56Co and 57Co produced in the explosion, or the time at which freeze-out sets in). We propose a new, late-time Phillips-like correlation between the stretch of the SNe and the shape of their late-time light curves, which we parametrize as the difference between their pseudo-bolometric luminosities at 600 and 900 days: ΔL900 = log(L600/L900). This model-independent correlation provides a new way to test which physical process lies behind the slow-down of SN Ia light curves >900 days after explosion, and, ultimately, fresh constraints on the various SN Ia progenitor and explosion models.