We perform a comprehensive study of the Higgs potential of the two Higgs doublet model extended by a real triplet scalar field [Formula: see text]. This model, dubbed [Formula: see text], has a rich Higgs spectrum consisting of three CP-even Higgs [Formula: see text], one CP-odd [Formula: see text] and two pairs of charged Higgs [Formula: see text]. First, we determine the perturbative unitarity constraints and a set of nontrivial conditions for the boundedness from below (BFB). Then we derive the Veltman conditions by considering the quadratic divergencies of Higgs boson self-energies in [Formula: see text]. 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 [Formula: see text] is then performed on the physical scalars of [Formula: see text]. 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 [Formula: see text], essentially when naturalness is involved. For [Formula: see text] scenario, our results place higher limits on the bounds of all scalar masses, and show that the pairs [Formula: see text] and [Formula: see text] are nearly mass degenerate varying within the intervals [Formula: see text] GeV and [Formula: see text] GeV, respectively. When [Formula: see text] 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: [Formula: see text] GeV for [Formula: see text]; [Formula: see text] GeV for ([Formula: see text], [Formula: see text]) and [Formula: see text] GeV for ([Formula: see text], [Formula: see text]). Finally, we find that the [Formula: see text] and [Formula: see text] Higgs decay modes are generally correlated if [Formula: see text] lies within the reduced intervals [Formula: see text] and [Formula: see text] parameter is frozen around [Formula: see text] ([Formula: see text]) for [Formula: see text] ([Formula: see text]).