  Traditional channel coding with feedback constructs and transmits a codeword
only after all message bits are available at the transmitter. This paper joins
Guo & Kostina and Lalitha et. al. in developing approaches for causal (or
progressive) encoding, where the transmitter may begin transmitting codeword
symbols as soon as the first message bit arrives. Building on the work of
Horstein, Shayevitz and Feder, and Naghshvar et. al., this paper extends our
previous computationally efficient systematic algorithm for traditional
posterior matching to produce a four-phase encoder that progressively encodes
using only the message bits causally available. Systematic codes work well with
posterior matching on a channel with feedback, and they provide an immediate
benefit when causal encoding is employed instead of traditional encoding. Our
algorithm captures additional gains in the interesting region where the
transmission rate mu is higher than the rate lambda at which message bits
become available. In this region, transmission of additional symbols beyond
systematic bits, before a traditional encoder would have begun transmission,
further improves performance
