A strong invariance principle is established for random fields which satisfy dependence conditions more general than positive or negative association. We use the approach of Csörg\H{o} and Révész applied recently by Balan to associated random fields. The key step in our proof combines new moment and maximal inequalities, established by the authors for partial sums of multiindexed random variables, with the estimate of the convergence rate in the CLT for random fields under consideration.