We designed and realized real-time pulse amplitude modulation-4 (PAM-4) digital signal processing (DSP) including forward error correction (FEC) for a C-band inter-datacenter network. The PAM-4 DSP is intended to compensate for chromatic dispersion and provide dispersion tolerance. A decision feedback equalizer (DFE) and maximum likelihood sequence equalizer (MLSE) were employed for the dispersion compensation. A low-density parity check (LDPC) code was used to increase coding gain. The soft-decision Viterbi algorithm (SOVA) was adopted to provide probabilistic information to the LDPC code. For implementation in a real-time field programmable gate arrays (FPGAs), we employed fully parallelized structures. In the design, three LDPC cores were operated in parallel, and the equalizers were also operated with 128 PAM-4 symbols. With the DSP, we empirically proved the feasibility of 25 km transmission without error-floor sign, corresponding to a dispersion compensation capacity of 425 ps/nm. We confirmed 35 km ∼ 85 km error-free transmission for inter-datacenter network.