  In this paper, we report on the detection of RRab stars with
quasi-identical-shape light curves but period differences as large as
$0.05-0.21$ d using the Galactic bulge data of the OGLE-IV survey. We have
examined stars with shorter periods than the Oosterhoff~I ridge of the bulge.
These stars generally have smaller amplitudes and larger Fourier
phase-differences than the typical bulge RRab stars have at the same period.
Many of these "anomalous" stars have good-quality light curves without any sign
of the Blazhko modulation. Examining their Fourier parameters revealed that
several of these stars show very similar light-curve to the typical bulge RR
Lyrae. We found hundreds of quasi-identical-shape light-curve pairs with
different periods between the "anomalous"- and the "normal"-position RRab stars
based on the OGLE $I$-band data. The OGLE $V$-band, and the archive VVV and
MACHO surveys $K_s$-, $b$- and $r$-band data of these stars were also checked
for light-curve-shape similarity. Finally, 149 pairs with identical-shape light
curves in each available photometric band were identified. Calculating the
physical properties of the variables using empirical formulae, on average,
$-0.5$~dex, $-0.13$~mag, 0.67, and 165~K differences between the [Fe/H], $M_V$,
$R/R_\odot$, and $T_{\mathrm{eff}}$ values of the members of the pairs are
derived, being the short-period stars less metal-poor, fainter, smaller and
hotter than the long-period variables. To explain the existence of variables
with different physical properties and pulsation periods but with
identical-shape light curves is a challenging task for modelling.
