The \^G Infrared Search for Extraterrestrial Civilizations with Large
  Energy Supplies. III. The Reddest Extended Sources in WISE
  Nearby Type III (galaxy-spanning) Kardashev supercivilizations would have
high mid-infrared (MIR) luminosities. We have used the Wide-field Infrared
Survey Explorer (WISE) to survey ~$1 \times 10^5$ galaxies for extreme MIR
emission, $10^3$ times more galaxies than the only previous such search. We
have calibrated the WISE All-sky Catalog pipeline products to improve its
photometry for extended sources. We present 563 extended sources with $|b| \ge
10$ and red MIR colors, having visually vetted them to remove artifacts. No
galaxies in our sample host an alien civilization reprocessing more than 85% of
its starlight into the MIR, and only 50 galaxies, including Arp 220, have MIR
luminosities consistent with >50% reprocessing. Ninety of these (likely)
extragalactic sources have little literature presence; in most cases they are
likely barely resolved galaxies or pairs of galaxies undergoing large amounts
of star formation. Five are new to science and deserve further study. The Be
star 48 Librae sits within a MIR nebula, and we suggest that it may be creating
dust. WISE, 2MASS, and Spitzer imagery shows that IRAS 04287+6444 is consistent
with a previously unnoticed, heavily extinguished cluster of young stellar
objects. We identify five "passive" (i.e. red) spiral galaxies with unusually
high MIR and low NUV luminosity. We search a set of optically "dark" HI
galaxies for MIR emission, and find none. These 90 poorly understood sources
and five anomalous passive spirals deserve follow-up via both SETI and
conventional astrophysics.
