  The recent measurement by WMAP of a large electron scattering optical depth
tau_e = 0.17 +- 0.04 is consistent with a simple model of reionization in which
the intergalactic medium (IGM) is ionized at redshift z ~ 15, and remains
highly ionized thereafter. Here, we show that existing measurements of the IGM
temperature from the Lyman-alpha forest at z ~ 2 - 4 rule out this ``vanilla''
model. Under reasonable assumptions about the ionizing spectrum, as long as the
universe is reionized before z = 10, and remains highly ionized thereafter, the
IGM reaches an asymptotic thermal state which is too cold compared to
observations. To simultaneously satisfy the CMB and forest constraints, the
reionization history must be complex: reionization begins early at z >~ 15, but
there must have been significant (order unity) changes in fractions of neutral
hydrogen and/or helium at 6 < z < 10, and/or singly ionized helium at 4 < z <
10. We describe a physically motivated reionization model that satisfies all
current observations. We also explore the impact of a stochastic reionization
history and show that a late epoch of (HeII --> HeIII) reionization induces a
significant scatter in the IGM temperature, but the scatter diminishes with
time quickly. Finally, we provide an analytic formula for the thermal
asymptote, and discuss possible additional heating mechanisms that might evade
our constraints.
