  Using vector magnetograms from the HMI/SDO and a magnetic connectivity-based
method, we calculate the instantaneous relative magnetic helicity and free
magnetic energy budgets for several days in two solar active regions (ARs),
AR11890 and AR11618, both with complex photospheric magnetic field
configurations. The ARs produced several major eruptive flares while their
photospheric magnetic field exhibited primarily flux decay in AR11890 and
primarily flux emergence in AR11618. Throughout much of their evolution both
ARs featured substantial budgets of free magnetic energy and of both positive
and negative helicity. In fact, the imbalance between the signed components of
their helicity was as low as in the quiet Sun and their net helicity eventually
changed sign 14-19 hours after their last major flare. Despite such
incoherence, the eruptions occurred at times of net helicity peaks that were
co-temporal with peaks in the free magnetic energy. The losses associated with
the eruptive flares in the normalized free magnetic energy were in the range
10-60%. For the helicity, changes ranged from 25% to the removal of the entire
excess helicity of the prevailing sign, leading a roughly zero net helicity,
but with significant equal and opposite budgets of both helicity senses. The
removal of the slowly varying background component of the free energy and
helicity timeseries revealed that all eruption-related peaks of both quantities
exceeded the 2$\sigma$ levels of their detrended timeseries. There was no
eruption when only one or none of these quantities exceeded its 2$\sigma$
level. Our results indicate that differently evolving ARs may produce major
eruptive flares even when, in addition to the accumulation of significant free
magnetic energy budgets, they accumulate large amounts of both negative and
positive helicity without a strong dominance of one handedness over the other.
