In conventional dual-energy systems, two transmission measurements with distinct spectral characteristics are performed. These measurements are used to obtain the line integrals of two basis material densities. Usually, the measurement process is such that the two measured signals can be treated as independent and therefore uncorrelated. Recently, however, a readout system for x-ray detectors has been introduced for which this is no longer the case. The readout electronics is designed to obtain simultaneous measurements of the total number of photons N and the total energy E they deposit in the sensor material. Practically, this is realized by a signal replication and separate counting and integrating processing units. Since the quantities N and E are (electronically) derived from one and the same physical sensor signal, they are statistically correlated. Nevertheless, the pair N and E can be used to perform a dual-energy processing following the well-known approach by Alvarez and Macovski. Formally, this means that N is to be identified with the first dual-energy measurement M1 and E with the second measurement M2. In the presence of input correlations between M1 = N and M2 = E, however, the corresponding analytic expressions for the basis image noise have to be modified. The main observation made in this paper is that for positively correlated data, as is the case for the simultaneous counting and integrating device mentioned above, the basis image noise is suppressed through the influence of the covariance between the two signals. We extend the previously published relations for the basis image noise to the case where the original measurements are not independent and illustrate the importance of the input correlations by comparing dual-energy basis image noise resulting from the device mentioned above and a device measuring the photon numbers and the deposited energies consecutively.