A technique termed gradual multifractal reconstruction (GMR) is formulated. A continuum is deined from a signal that preserves the pointwise H lder exponent (multifractal) structure of a signal but randomises the locations of the original data values with respect to this (φ = 0), to the original signal itself (φ = 1). We demonstrate that this continuum may be populated with synthetic time series by undertaking selective randomisation of wavelet phases using a dual-tree complex wavelet transform. That is, the φ = 0 end of the continuum is realised using the recently proposed iterated, amplitude adjusted wavelet transform algorithm [Keylock, C. J. 2017. Phys. Rev. E 95, 032123] that fully randomises the wavelet phases. This is extended to the GMR formulation by selective phase randomisation depending on whether or not the wavelet coeicient amplitudes exceeds a threshold criterion. An econophysics application of the technique is presented. The relation between the normalised log-returns and their H lder exponents for the daily returns of eight inancial indices are compared. One particularly noticeable result is the change for the two american indices (NASDAQ 100 and S&P 500) from a non-signiicant to a strongly signiicant (as determined using GMR) cross-correlation between the returns and their H lder exponents from before the 2008 crash to afterwards. This is also relected in the skewness of the phase diference distributions, which exhibit a geographical structure, with asian markets not exhibiting signiicant skewness in contrast to those from elsewhere globally.