  We perform a comprehensive study of The Higgs potential of the two Higgs
doublet model extended by a real triplet scalar field $\Delta$. This model,
dubbed $2\mathcal{HDM+T}$, has a rich Higgs spectrum consisting of three
CP-even Higgs $h_{1,2,3}$, one CP-odd $A_0$ and two pairs of charged Higgs
$H^\pm_{1,2}$. First, we determine the perturbative unitarity constraints and a
set of non trivial conditions for the boundedness from below (BFB). Then we
derive the Veltman conditions by considering the quadratic divergencies of
Higgs boson self energies in $2\mathcal{HDM+T}$. We find that the parameter
space is severely delimited by these theoretical constraints, as well as
experimental exclusion limits and Higgs signal rate measurements at LEP and
LHC. Using HiggsBounds-5.3.2beta and HiggSignals-2.2.3beta public codes an
exclusion test at $2\sigma$ is then performed on the physical scalars of
$2\mathcal{HDM+T}$. Our analysis provides a clear insight on the nonstandard
scalar masses, showing that the allowed ranges are strongly sensitive to the
sign of mixing angle $\alpha_1$, essentially when naturalness is involved. For
$\alpha_1 < 0$ scenario, our results place higher limits on the bounds of all
scalar masses, and show that the pairs $(h_2, H_1^\pm)$ and $(h_3, H_2^\pm)$
are nearly mass degenerate varying within the intervals $[130\,,\,246]$~GeV and
$[160\,,\,335]$~GeV respectively. When $\alpha_1$ turns positive, we show that
consistency with theoretical constraints and current LHC data, essentially on
the diphoton decay channel, favors Higgs masses varying within wide allowed
ranges: $[153\,,\,973]$~GeV for $m_{A_0}$; $[151\,,\,928]$~GeV for ($m_{h_2}$,
$m_{H_1^\pm}$) and $[186\,,\,979]$~GeV for ($m_{h_3}$, $m_{H_2^\pm}$). Finally,
we find that the $\gamma \gamma$ and $Z\gamma$ Higgs decay modes are generally
correlated.
