  Evidence that the multiple populations (MPs) are common properties of
globular clusters (GCs) is accumulated over the past decades from clusters in
the Milky Way and in its satellites. This finding has revived GC research, and
suggested that their formation at high redshift must have been a much-more
complex phenomenon than imagined before. However, most information on MPs is
limited to nearby GCs. The main limitation is that most studies on MPs rely on
resolved stars, facing a major challenge to investigate the MP phenomenon in
distant galaxies. Here we search for integrated colors of old GCs that are
sensitive to the multiple-population phenomenon. To do this, we exploit
integrated magnitudes of simulated GCs with MPs, and multi-band Hubble Space
Telescope photometry of 56 Galactic GCs, where MPs are widely studied, and
characterized as part of the UV Legacy Survey of Galactic GCs. We find that
both integrated $C_{\rm F275W,F336W,F438W}$ and $m_{\rm F275W}-m_{\rm F814W}$
colors strongly correlate with the iron abundance of the host GC. In second
order, the pseudo two-color diagram built with these integrated colors is
sensitive to the MP phenomenon. In particular, once removed the dependence from
cluster metallicity, the color residuals depend on the maximum internal helium
variation within GCs and on the fraction of second-generation stars. This
diagram, which we define here for Galactic GCs, has the potential of detecting
and characterizing MPs from integrated photometry of old GCs, thus providing
the possibility to extend their investigation outside the Local Group.
