We study the topology of real analytic maps in a neighborhood of a (possibly non-isolated) critical point. We prove fibration theorems à la Milnor for real analytic maps with non-isolated critical values. Here we study the situation for maps with arbitrary critical set. We use the concept of d-regularity introduced in an earlier paper for maps with an isolated critical value. We prove that this is the key point for the existence of a Milnor fibration on the sphere in the general setting. Plenty of examples are discussed along the text, particularly the interesting family of functions $$(f,g):{\mathbb {R}}^n \rightarrow {\mathbb {R}}^2$$
(
f
,
g
)
:
R
n
→
R
2
of the type $$\begin{aligned} (f,g) = \left( \sum _{i=1}^n a_i x_i^p, \sum _{i=1}^n b_i x_i^q \right) , \end{aligned}$$
(
f
,
g
)
=
∑
i
=
1
n
a
i
x
i
p
,
∑
i
=
1
n
b
i
x
i
q
,
where $$a_i, b_i \in {\mathbb {R}}$$
a
i
,
b
i
∈
R
are constants in generic position and $$p,q \ge 2$$
p
,
q
≥
2
are integers.