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 $2mathcal{HDM+T}$, 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 $2mathcal{HDM+T}$. 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 $2sigma$ is then performed on the physical scalars of $2mathcal{HDM+T}$. 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 $Zgamma$ Higgs decay modes are generally correlated.
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