  An analysis is presented of events containing jets including at least one
$b$-tagged jet, sizable missing transverse momentum, and at least two charged
leptons including a pair of the same electric charge, with the scalar sum of
the jet and lepton transverse momenta being large. Standard Model processes
rarely produce these final states, but several models of physics beyond the
Standard Model predict an enhanced production rate of such events. Specific
models with this feature are considered here: vector-like $T$, $B$, and
$T_{5/3}$ quark pair production, and four top quark production under three
scenarios (Standard Model, contact interaction, and extra-dimensions). A data
sample of 3.2 fb$^{-1}$ of $pp$ collisions at a center-of-mass energy of
$\sqrt{s}$=13 TeV recorded by the ATLAS detector at the Large Hadron Collider
is used in this analysis. Several signal regions are defined, in which the
consistency between the data yield and the background-only hypothesis is
checked, and 95% confidence level limits are set on various signal models. The
focus here is on models yielding signatures with four top quarks.
