To my knowledge, $ \mathbb{R} \subset \mathbb{C}$ and not vice-versa.
Then I don't understand why I get more real-valued solutions with Reals, than with Complexes:
$Assumptions =
a > 0 && a \[Element] Reals && b \[Element] Reals &&
t \[Element] Reals && t >= 0 && c1 \[Element] Reals &&
c2 \[Element] Reals && c \[Element] Reals && B \[Element] Reals &&
c2 > 0 && c1 < 0
Solve[Log[Abs[(B - c2)/(B - c1)]] == (a t - c) (c2 - c1), B, Reals] // Simplify
Solve[Log[Abs[(B - c2)/(B - c1)]] == (a t - c) (c2 - c1), B, Complexes] // Simplify